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Institut fur Festkorperphysik E13

Technische Universitat Munchen

Nuclear Resonant Scattering

for the Study of Dynamics

of Viscous Liquids and Glasses

Ilia Sergueev

Vollstandiger Abdruck der von

der Fakultat fur Physik der Technischen Universitat Munchen zur Erlangung

des akademischen Grades eines

Doktors der Naturwissenschaften (Dr. rer. nat.)

genehmigten Dissertation.

Vorsitzender: Univ.-Prof. Dr. P. Ring

Prufer der Dissertation: 1. Univ.-Prof. Dr. W. Petry

2. Univ.-Prof. Dr. F. E. Wagner, i.R.

Die Dissertation wurde am 09.12.2003 bei der

Technischen Universitat Munchen eingereicht und durch die

Fakultat fur Physik am 04.02.2004 angenommen.

Abstract

Nuclear resonant scattering of synchrotron radiation on molecular probes has been em-

ployed to study the dynamics of a glass former. Incoherent and coherent scattering have

been investigated in theory and applied in experiment in the static case and in the presence

of relaxation. The influence of translational and spin relaxation in the pico-to-microsecond

time range on relevant observables has been characterized. The measured temperature

dependence of these observables gives then information about the liquid-to-glass transi-

tion. In particular the combination of coherent and incoherent scattering allowed here to

separate translational and rotational dynamics.

Zusammenfassung

Kernresonante Streuung von Synchrotronstrahlung an molekularen Sonden wurde einge-

setzt um die Dynamik eines Glasbildners zu untersuchen. Inkoharente und koharente

Streuung wurden im statischen Fall und bei Relaxation theoretisch untersucht und exper-

imentell angewandt. Der Einfluß von Translations- und Spin-Relaxation im Zeitbereich

von Piko- bis Mikrosekunden auf wichtige Meßgroßen wurde charakterisiert. Das exper-

imentell bestimmte Temperaturverhalten dieser Großen ermoglicht dann Aussagen uber

den Flussig-Glas-Ubergang. Insbesondere erlaubte hier die Kombination koharenter und

inkoharenter Streuung Translations- und Rotationsdynamik zu trennen.

iii

List of Abbreviations

APD Avalanche Photo-Diode

DB Dynamical Beat

DBP Dibutyl Phthalate

DOS Density of States

DS Dielectric Spectroscopy

EFG Electric Field Gradient

ESRF European Synchrotron Radiation Facility

FC Ferrocene

FJM Finite Jump Model

HRM High Resolution Monochromator

LGT Liquid-to-Glass Transition.

MCT Mode Coupling Theory

MS Mossbauer Spectroscopy

NFS Nuclear Forward Scattering

NIS Nuclear Inelastic Scattering

NMR Nuclear Magnetic Resonance

NQR Nuclear Quadrupole Resonance

NRS Nuclear Resonant Scattering

PM Premonochromator

QB Quantum Beat

v

vi

RDM Rotational Diffusion Model

SCM Strong Collision Model

SR Synchrotron Radiation

SRPAC Synchrotron Radiation based Perturbed Angular Correlation

TDI Time Domain Interferometry

TDPAC Time Differential Perturbed Angular Correlation

VFT Vogel-Fulcher-Tammann equation

Contents

1 Introduction 1

2 Relaxation in glass-forming liquids 5

2.1 General aspects of the liquid-to-glass transition (LGT) . . . . . . . . . . . 5

2.2 Temperature dependence of LGT dynamics . . . . . . . . . . . . . . . . . . 9

2.3 Time dependence of the relaxation function . . . . . . . . . . . . . . . . . 11

2.4 Description of the LGT by the mode-coupling theory . . . . . . . . . . . . 12

2.5 Slow β relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Nuclear forward scattering (NFS) of synchrotron radiation (SR) 19

3.1 The Mossbauer effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Principles of NFS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Theory of NFS in the static case . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3.1 Single resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3.2 Hyperfine splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4 Influence of spatial dynamics on NFS . . . . . . . . . . . . . . . . . . . . . 26

3.4.1 General formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4.2 Debye relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.4.3 Kohlrausch relaxation . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.5 Time domain interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4 SR-based perturbed angular correlation (SRPAC) 37

4.1 Spatially coherent versus incoherent nuclear resonant scattering . . . . . . 37

4.2 Principle of SRPAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.3 Theory of SRPAC in the static case . . . . . . . . . . . . . . . . . . . . . . 42

4.4 Influence of spin dynamics on SRPAC . . . . . . . . . . . . . . . . . . . . . 46

4.4.1 Slow relaxation regime . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.4.2 Fast relaxation regime . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.4.3 Intermediate relaxation regime . . . . . . . . . . . . . . . . . . . . . 54

4.4.4 Short-time dynamics in restricted anglular range . . . . . . . . . . . 54

vii

viii Contents

4.5 Influence of spatial and spin dynamics on NFS . . . . . . . . . . . . . . . . 56

4.5.1 Influence of spin dynamics . . . . . . . . . . . . . . . . . . . . . . . 56

4.5.2 Influence of spin and spatial dynamics . . . . . . . . . . . . . . . . 58

5 Experimental 61

5.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.2 Methodical aspects of SRPAC . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.2.1 Dependence on the geometry of the experiment . . . . . . . . . . . 63

5.2.2 Contributions to 4π scattering produced by NFS . . . . . . . . . . . 66

6 Study of LGT dynamics by NFS and SRPAC 71

6.1 Previous studies of the LGT by Mossbauer spectroscopy . . . . . . . . . . 71

6.2 The sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.3 Experiment by SRPAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.4 Experiment by NFS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.5 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.5.1 Fast dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.5.2 Slow dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.6 Stretched exponential relaxation . . . . . . . . . . . . . . . . . . . . . . . . 97

7 Study of LGT dynamics in restricted geometry by NFS 101

7.1 Dynamics of viscous liquids in restricted geometry . . . . . . . . . . . . . . 101

7.2 The sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7.3 Experiment by NFS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

8 Conclusion and Outlook 111

A Calculation of the NFS amplitude 115

B Electric quadrupole interaction 119

C Calculation of the SRPAC intensity 121

D Calculation of G22(t) by the eigensystem method 125

E Calculation of G22(t) by the resolvent method 131

Chapter 1

Introduction

The understanding of the formation of glasses by slowing down the dynamics of liquids

is currently seen as one of the major challenges in condensed matter physics. Moreover,

the glassy products formed by this process provide a large variety of important materials

which are widely used in nowadays technologies. The dynamics of the liquid-to-glass

transition extends over a very wide time window from short time dynamics in the 0.1ps

domain to infinitely long times, i.e. in practical terms over more than 14 decades. Different

experimental techniques are employed to cover such a large time window. One of them is

Mossbauer spectroscopy.

Mossbauer Spectroscopy (MS) is a high-resolution spectroscopic method widespread

in solid state physics. It relies on recoilless emission and absorption of nuclear resonant

γ-radiation, the probability of which is given by the Lamb-Mossbauer factor. MS is

sensitive to dynamics in a ns-µs time window and on a sub-Angstrom to Angstrom length

scale. These features make it a powerful tool to study the liquid-to-glass transition. The

method was successfully applied to study dynamics in soft matter, e.g. in organic glasses,

polymers and biological compounds.

After synchrotron radiation (SR) was introduced as a new tool for scientific research,

it was suggested to use it for the excitation of Mossbauer nuclei. The first successful

nuclear resonant scattering experiment using synchrotron radiation was performed in

1985 in Bragg reflection. Later on, the method of Nuclear Forward Scattering (NFS) was

developed which is the coherent analogue of MS on the time scale. It is particularly suited

to investigate hyperfine interactions and dynamics. The next step in the development

was nuclear inelastic scattering where information about the density of states is directly

obtained. Both methods have become highly attractive nowadays due to the excellent,

continuously improving characteristics of the available SR sources. Intense X-ray beams

with extreme brilliance and almost completely linear polarization allow us to investigate

tiny sample volumes and samples under special conditions. Due to the possible small

1

2 Chapter 1. Introduction

sample sizes combinations of high pressures, low temperatures and high magnetic fields

can be realized, which enable us to investigate the magnetic and electronic properties of

solid state matter down to nanoparticles.

However, both MS and NFS are restricted to investigations of materials with non-

vanishing Lamb-Mossbauer factor. In particular, the liquid-to-glass transition can be ap-

proached by these methods only from the glassy state. This restriction can be overcome

by a new method, Synchrotron Radiation based Perturbed Angular Correlation (SRPAC).

SRPAC relies on spatially incoherent nuclear resonant scattering of SR and can be consid-

ered as a scattering variant of Time Differential Perturbed Angular Correlation (TDPAC).

However, in SRPAC the investigated nuclear level is populated not ’from above’ (via a

cascade as in TDPAC) but ’from below’, i.e. from the ground state. The absence of the

cascade increases the amount of isotopes which can be investigated. In particular, SRPAC

can be applied to the Mossbauer isotopes like 57Fe, 119Sn, 61Ni.

Being single-nucleus scattering, SRPAC does not depend on recoilfree emission and

absorption, and not on translational nuclear motion. Therefore SRPAC allows one to

continue Mossbauer investigations of hyperfine interactions and rotational dynamics into

regions where the Lamb-Mossbauer factor vanishes, i.e. in very soft matter and viscous

liquids. It also opens the possibility to investigate high-energy Mossbauer transitions

which are otherwise often inaccessible at ambient temperatures due to a vanishing Lamb-

Mossbauer factor.

The main aim of this work was to develop the method of SRPAC and to apply the

combination of SRPAC and NFS to study dynamics of viscous liquids. This aim includes

several tasks:

• Theoretical

– Formulation of SRPAC in the static case. In principle, the theory of angular

correlation can be used to describe SRPAC. However, a different way of forma-

tion of the investigated nuclear level and the linear polarization of SR require

modifications of the theory.

– Formulation of the influence of rotational dynamics on SRPAC for the 57Fe

nucleus. The influence of dynamics on angular correlation was studied theo-

retically very extensively in the 70’s in application to TDPAC. However, the

large development of the physics of viscous liquids during the last two decades

introduced new theoretical approaches, which have to be included. Dynami-

cal aspects of SRPAC are similar to those of nuclear magnetic resonance. The

models used in this method can be successfully applied to SRPAC. Further, the

spin quantum number of the used isotope can give the possibility to simplify

the general theory and sometimes to reduce its results to analytical expressions.

3

– Formulation of the influence of Kohlrausch relaxation on NFS. The essential

property of the dynamics of viscous liquids is the stretching of relaxation. It

is usually described by a Kohlrausch relaxation function. The influence of this

relaxation on NFS has to be analyzed.

• Experimental

– Methodical study of SRPAC. The experimental realization of SRPAC is com-

pletely different compared to TDPAC. A methodical experimental study of

SRPAC has to be performed and the optimal experimental setup for studying

dynamics has to be found.

– Use of SRPAC to study dynamics of the liquid-to-glass transition. Experiments

which show the perspectives and restrictions of SRPAC to study dynamics have

to be performed.

– Separation of dynamics into pure rotation and translation of the center of mass.

The combination of SRPAC which is sensitive only to the rotational molecu-

lar dynamics and NFS which is sensitive to both rotational and translational

molecular dynamics gives in principle the possibility to separate these two

types of dynamics. A comparison of the translational and rotational molecular

motions gives new information about the liquid-to-glass transition.

– Study of the liquid-to-glass transition in restricted geometry by NFS. The study

of liquid dynamics in confinement has attracted much interest nowadays. It is

believed that confinement may help to address fundamental questions about

bulk dynamics. The realization of such an experiment is presented in this work.

The arrangement of this work is the following. Chapter 1 gives a short introduction

to the physics of the liquid-to-glass transition. The main features are described and the

fundamental mode coupling theory is introduced. Chapter 2 describes NFS. The influence

of the relaxation driven by exponential and Kohlrausch relaxation functions on NFS is

described. Chapter 3 is dedicated to the theory of SRPAC. Here the static case and the

influence of dynamics are described. Additionally, the influence of molecular rotation

on NFS which is similar to that on SRPAC is shown. Chapter 4 is dedicated to the

description of the experimental setup and to methodical aspects of SRPAC. A study of

a glass forming liquid by a combination of SRPAC and NFS is presented in Chapter 5.

The obtained results are discussed. The influence of confinement on the liquid-to-glass

transition is presented in Chapter 6. Here the same glass forming liquid is used as in

the bulk sample. A difference between dynamics in confinement and in the bulk is found

and discussed. In Chapter 7 the results are summarized and perspectives of SRPAC are

discussed.

4 Chapter 1. Introduction

Chapter 2

Relaxation in glass-forming liquids

Glass, in the popular and basically correct conception, is a liquid that has lost its ability

to flow [Ang95]. However, while glasses form an extremely important class of materials

and the glass-forming ability is shared by many different substances, the phenomena

underlying the liquid-to-glass transition (LGT) and the nature of the glassy state are not

yet fully understood. The main questions that physicists try to solve are:

• Why do certain substances or solutions suddenly undergo a dramatic ”slowing down”

in the diffusive motions of their particles?

• Why do glasses not form a precisely ordered crystalline material, at some precisely

defined freezing point, like other, more ”normal” substances?

Those questions also motivated the experiments described in this thesis. A short review

will be given in this chapter, describing the following features: general aspects of the LGT,

temperature dependence of the dynamics of the LGT, time dependence of the relaxation

function, description of the LGT in the mode-coupling theory (MCT) and the slow β

relaxation.

2.1 General aspects of the liquid-to-glass transition

The usual way to obtain a glass is fast cooling (quenching) of a liquid. If no crystals are

formed during this process, the glassy state is entered when the liquid passes through the

liquid-to-glass transition, which is a range of temperatures over which the system ”falls out

of equilibrium” (In strict equilibrium thermodynamics the viscous liquid is already below

melting temperature in a non-equilibrium state). The clearest signature of the approach

to the LGT is a huge increase in the viscosity η. Whereas η ' 10−2 . . . 10−1 poise is a

typical value for a normal liquid, the glass transition temperature Tg is usually defined as

the temperature where η reaches 1013 poise [Ang95].

5

6 Chapter 2. Relaxation in glass-forming liquids

Figure 2.1: Different forms of heat capacity for liquid (l) and crystal (c) phases of several

glass formers [Ang95].

In a narrow temperature range around Tg, certain properties of the liquid such as

specific volume, enthalpy and heat capacity change abruptly. The dependence of the heat

capacity Cp on temperature is shown in Fig. 2.1 for several glass formers [Ang95]. It is

seen that the heat capacity changes from a liquid-like to a crystal-like value at the region

around Tg. This phenomenon can be used for another definition of Tg as the temperature

of a discontinuity of Cp. For slow cooling rates or typical observation time of 100−1000 s

both definitions give similar values for Tg. Therefore Tg is also called the caloric glass

transition temperature. The first conclusion from the behavior of the heat capacity would

be that there is a second-order phase transition at Tg. However, no discontinuity in the

behavior of the viscosity is observed at this temperature. To explain this contradiction

we consider qualitatively the dynamics of a liquid during cooling. A normal liquid can

be visualized as a large collection of tiny particles that are in a state of ceaseless violent

motion. Chaotic trajectories bring particles into collision with neighboring particles. On

the average, the collisions result in a reversal of the trajectories of the particles, so they

appear to be rattling in the cage formed by their neighbors with a characteristic time

τl. But sometimes changes of the relative positions occur and the particle jumps out

from the cage to another position that becomes the new center of the rattling motion.

Such a structural relaxation or diffusive motion occurs with a characteristic time τr. In

the normal liquid the rattling motion and the structural relaxation occur with similar

characteristic times. It is difficult to separate them, therefore it is reasonable to assume

that the particles are undergoing continuous diffusion.

2.1. General aspects of the liquid-to-glass transition (LGT) 7

As the temperature decreases below the melting temperature Tm, the liquid becomes

supercooled. The packing becomes more dense and the particle spends more time rattling

in the cage. Structural relaxation requires that an increasing number of neighbors has to

cooperate in order to enable the particle to jump out of the cage. While the characteristic

rattling time τl changes only slightly, the structural relaxation time τr increases drastically.

The essential feature of the LGT is that an enormous gap opens between τl and τr. The

difference between these two time scales can change by up to ten orders of magnitude in

the temperature range from 1.1 Tg to Tg.

The reaction of the substance to external stress, which is a value that is measured

in experiments, consists of two parts well separated in time: crystal-like reaction in time

τl and liquid-like reaction in time τr. Each experiment has a characteristic time scale,

given by the time te, to probe the system and to observe the results. For heat capacity

measurements, te may be of the order of 100 s. The result of the experiment depends on

the ratio between te and τr. If te is long compared with τr, then a liquid-like reaction will

be seen in the experiment, in the opposite case a crystal-like behavior will be observed.

From the point of view of heat capacity measurements we can say that Tg is defined as the

temperature where the structural relaxation time becomes larger than 100 s. Viscosity,

on the other hand, depends only on structural relaxation and can be observed as such

on any time scale. That is why no abrupt change of the temperature dependence of the

viscosity is observed in the region of Tg.

The main result of these considerations is that the LGT is not a phase transition but

rather a kinetic transition to the state where the system is not in equilibrium any more.

In statistical mechanics the idea of ergodicity is introduced for such systems. When a

system is in the thermodynamical equilibrium it is called ergodic, otherwise it is called

non-ergodic.

As one can see Tg is not only a characteristic feature of a substance but strongly de-

pends on the experiment. The question is, whether any temperature exists that separates

crystal-like and liquid-like states of the substance, thus being a characteristic property of

the substance. The results of this thesis may help to answer that question.

One of the possible candidates for such a universal temperature results from ther-

modynamic considerations. It is the Kauzmann temperature TK , which appears in the

behavior of the thermodynamical parameters of a substance, e.g. of the entropy. The

entropy of a supercooled liquid has a larger value than that of the corresponding crystal

state at the same temperature. The excess entropy, called the entropy of fusion, comes

from the additional freedom of motion in a liquid as compared to a crystal. The depen-

dence of the entropy of fusion on temperature is shown in Fig. 2.2. It decreases with

decreasing temperature. It is not possible to measure the equilibrium liquid entropy be-

low Tg, but the extrapolation shows that below some temperature the entropy of fusion


Figure 2.2: Difference between liquid and crystalline state entropy for lithium acetate as

a function of temperature [WA76].

becomes negative. The vibrational and rattling contribution to the entropy of a glass

and the vibrational contribution to the entropy of a crystal are nearly the same, and the

entropy of a glass can not be smaller than that of a crystal. As a consequence the entropy

must deviate significantly from the extrapolation line through data shown in Fig. 2.2.

The temperature where the entropy of fusion apparently becomes negative is called the

Kauzmann temperature TK [Kau48, Ang95].

Another possible universal temperature is obtained from kinetic considerations. It

is the crossover temperature Tc first introduced by Goldstein [Gol69] in 1969 and now

strongly reinforced by mode-coupling theory [GS92, Got99]. Tc is located in the super-

cooled region above Tg. When particles in a liquid are packed more closely with decreasing

temperature, the transport of a particle out of the cage becomes more and more difficult.

At the crossover temperature structural relaxation and rattling in the cage decouple and

seen from the time scale of rattling the particle is frozen inside the cage. For T < Tc

another mechanism of diffusion becomes important. It is the thermally activated hopping

that defines structural relaxation below Tc. One experimental evidence for the signifi-

cance of the crossover temperature is the decoupling of different modes of motion below

Tc. There is no reason for the activation energy for diffusion connected with hops of a

single particle to be the same as the activation energy for the viscosity, which is related to

the motion of many particles. Fig. 2.3 shows an example of such decoupling in the organic

glass former o-terphenyl. The translational diffusion coefficient, which is proportional to

the corresponding relaxation time, is shown together with the viscosity scaled to have the

same behavior in the normal liquid regime [FGSF92]. The crossover temperature Tc for

2.2. Temperature dependence of LGT dynamics 9

Tc Tg

Figure 2.3: Translational diffusion coefficient measured by various techniques (•, M, ¤, M)

and the inverse viscosity (solid line) as functions of temperature for o-terphenyl [FGSF92];

Tc = 290 K, Tg = 240 K.

this system was determined independently by neutron scattering [PBF+91]. The viscosity

and the translational diffusion constant coincide in the high temperature region, so they

are governed by the same transport mechanism. But near Tc they decouple, and at Tg their

difference becomes two orders of magnitude. The same phenomenon was demonstrated

recently in metallic glasses [ZRFM03].

2.2 Temperature dependence of LGT dynamics

The main parameter of the LGT dynamics is the structural relaxation time τr which is

associated with transport or relaxation processes. The viscosity η is a measure of the

liquid response to a suddenly imposed shear stress and is related to the corresponding

relaxation time by the so-called Maxwell equation [CLH+97]:

η = G∞τr (2.1)

where G∞ is the high-frequency shear modulus, an elastic property of a liquid. There-

fore the temperature dependences of viscosity and relaxation time are the same for an

appropriate scaling factor G∞.

The easiest way to introduce a temperature dependence of relaxation is to consider

relaxation as a thermally activated process. The particle can jump from one site to

another if its thermal energy exceeds the energy barrier between two sites. The simplest

approach to describe the relaxation time for such thermally activated process is

τr = τl exp(∆E/kBT ) (2.2)


Figure 2.4: Tg scaled Arrhenius plots for viscosities of different glass-forming liquids

showing the spread of the data between the strong and fragile extremes [ANM+00].

where ∆E means the barrier energy, kB is the Boltzmann factor and τl is the characteristic

relaxation time in a normal liquid. This is the universal Arrhenius law of diffusion that

is used in many thermodynamic applications.

As it was observed in many glass-forming liquids, the temperature dependence of

the viscosity deviates from the Arrhenius behavior as it is seen in Fig. 2.4 [ANM+00]

where all substances are scaled in temperature by Tg (so called Angell plot of viscosity).

Open network liquids like SiO2 show Arrhenius variation of the viscosity (or structural

relaxation time) between Tg and the high temperature limit and provide the ”strong”

liquid extreme of the pattern. Other glass forming liquids, characterized by simple non-

directional Coulomb attractions or by van der Waals interactions, mainly organic liquids,

provide the other extreme, ”fragile” liquids. In fragile liquids, the viscosity varies in a

strongly non-Arrhenius fashion between the high and low limits. The strong/fragile-liquid

pattern is used as a basis to classify glass-forming liquids.

The most frequently applied equation to describe the deviation of the temperature de-

pendence of the structural relaxation time from an Arrhenius law is the phenomenological

Vogel-Fulcher-Tammann(VFT) equation

τr = τl exp[DT0/(T − T0)] (2.3)

where D and T0 are phenomenological parameters. The VFT equation often fits τr over

a considerably wide temperature range, but in almost no case over the full temperature

2.3. Time dependence of the relaxation function 11

range from Tg to Tb (boiling temperature). Choosing an appropriate fitting range T0 can

be found to be within 2% of the Kauzmann temperature [Ang91]. The parameter D

controls how closely the system obeys the Arrhenius law which corresponds to T0 → 0

and DT0 → ∆E/kB. D for a set of different glass-forming liquids is given in [BNAP93], it

varies from 150 for the strongest network liquids SiO2 and GeO2 to 1.5 for polymer melts.

MCT derives another equation to describe the transport properties or the structural

relaxation time

τr = A

(

T − Tc

Tc

)−γ

(2.4)

where γ will be defined in the section 2.4. This equation is applied for liquids in the highly

fluid to moderately viscous regime near the crossover temperature Tc (see section 2.4).

2.3 Time dependence of the relaxation function

Next in importance to the characteristic time of structural relaxation τr is the func-

tion which describes the relaxation process. Usually in spectroscopic methods the self-

correlation function Fs(t) is used for this purpose:

Fs(t) = 〈A(0)A(t)〉 (2.5)

It describes the relaxation of the dynamical variable A(t) (a quantum operator in general)

between time zero and a later time t. Physically, Fs(t) measures the relaxation time

over which the variable A retains its own memory until this memory is averaged out by

statistical randomness. The position of a molecule or its orientation can be chosen as A.

In the simplest case the structural relaxation function is an exponential, and a unique

time τr characterizes the process. In viscous liquids and glasses, however, exponential

responses are stretched to longer times, and the process has to be characterized by a

more complex function. The most common way to describe the time dependence of the

structural relaxation is the Kohlrausch function [Koh47]

Fs(t) = f exp[

−(t/τr)β]

(2.6)

where β is a stretching parameter with 0 < β 6 1 and f is a scaling factor that reflects

the presence of faster processes which precede the structural relaxation. This function is

empirical, and the physical meaning of β and its correlation with D are under debate.

Another important question is whether β is a universal feature of the substance and

whether it stays constant with temperature or not. The time-temperature superposition

principle, introduced by Ferry [Fer50] and reinforced by MCT, says that changing T or

other control parameters like density results in changing only the scale τr of the structural


relaxation process but not its functional form. Then the relaxation functions for different

temperatures can be collapsed to one master curve

FT (t) = Fs(t/τr(T )) (2.7)

For the case of a Kohlrausch function, that implies that β stays constant with changing

temperature.

From another point of view, there are some experiments showing that β increases

monotonically with increasing temperature and approaches unity in the normal liquid

regime [DWN+90, ANM+00].

2.4 Description of the LGT by the mode-coupling

theory

Mode coupling theory, an outgrowth of the kinetic theory of liquids, was first proposed

in 1984 [BGS84, Leu84] to explain the LGT. MCT is a fundamental physical theory that

describes the behavior of glass-forming liquids and explains many properties of the LGT

(for a review see [GS92, Got99]).

As it was mentioned before, the motion of a particle in a liquid can be separated into

two parts: the rattling inside the cage and the diffusion out of the cage (structural relax-

ation). This separation is defined by the time scale. At a short time scale a free particle

rattles in a cage with the characteristic time of the liquid dynamics τl ' 10−12 . . . 10−14 s.

This one-particle process is described by an exponential decay of the relaxation function.

Coming from one particle to clusters of particles, one can consider their dynamics in the

cage of their neighbors. This is again a local process, since there is no transport of par-

ticles out of the cage. Such cluster dynamics is presented as a long-time, stretched tail

of the one-particle rattling. This process is called fast β relaxation and is defined by the

time τβ. The diffusion out of the cage requires cooperative motion of the neighbors of

the particle. It results in the structural α relaxation with characteristic time τr. This

process is not exponential and, as it was mentioned before, can empirically be described

by a Kohlrausch stretched-exponential function. As a result the relaxation function con-

sists of two steps: the microscopical relaxation towards the fast β relaxation tail and the

structural α relaxation (see Fig. 2.5).

In the normal liquid regime, no cages can be formed during time τl and α and fast

β relaxations have the same time scale. With decreasing temperature (increasing the

packing density of particles) time scales start to diverge from each other. There is a

critical temperature Tc, and that is the main point of the basic version of MCT, where

this divergence becomes infinite. The particles would be frozen in their cages. Therefore

2.4. Description of the LGT by the mode-coupling theory 13

Figure 2.5: Relaxation function as function of log10 t calculated for different packing

fractions φ [FFG+97] which can be associated with temperatures T [FFG+97]. The thick

curve labels the critical fraction φc. Number n shows deviation from φc from both sides:

φ = φc(1 ± 10−n/3).

at T 6 Tc the relaxation function decays not to zero but to some finite value defined by

fast β relaxation (see Fig. 2.5). This value is called the glass form factor or non-ergodicity

parameter f . It can be identified with the Debye-Waller factor fQ or the Lamb-Mossbauer

factor fLM . Above but near the critical temperature the step connected with α-relaxation

exists, and it is well separated from the step connected with fast β-relaxation by a plateau

of height f with a width that decreases with increasing temperature. Here f can be

identified with the Debye-Waller factor as measured on inelastic neutron spectrometers

with an energy resolution in the order of µeVs or with the Lamb-Mossbauer factor in

Mossbauer spectroscopy since it does not take into account the long-time α relaxation.

The temperature behavior of f near Tc depends on the deviation of the temperature from

Tc and can be described using σ = (Tc − T )/Tc as

f =

f c + h√σ T < Tc

f c T > Tc

(2.8)

where f c is a critical glass form factor defined mainly by the experimental scattering vector

and h is a critical amplitude. An experimental observation of the temperature behavior

of the Debye-Waller factor is shown in Fig. 2.6 [TSWF97]. The square root anomaly is

seen for various values of the scattering vector at the same temperature Tc ≈ 290 K.

MCT predicts a fractal time dependence of the relaxation function in the region close


Figure 2.6: Effective Debye-Waller factors fQ of o-terphenyl as functions of temperature T

for various wave vectors Q by coherent neutron scattering spectroscopy [TSWF97]. The

curves indicate fits leading to a critical temperature Tc ≈ 290 K.

to the plateau for both steps of relaxation:

Fs(t) ≈ f + h

(

t

τβ

)−a

for τ0 ¿ t¿ τr (2.9)

Fs(t) ≈ f − h

(

t

τr

)b

for τβ ¿ t and T > Tc (2.10)

where a and b are critical exponents. They are determined by the system-dependent

parameter λ as

λ =Γ2(1 − a)

Γ(1 − 2a)=

Γ2(1 + b)

Γ(1 + 2b)(2.11)

where Γ denotes the gamma function. One can see that the time-temperature superpo-

sition principle is valid for both steps of relaxation. Relaxation functions for different

temperatures can be collapsed to one master curve for fast β relaxation and separately

for α relaxation. The characteristic times for both steps of relaxation are determined by

the relative deviation σ from Tc and by the critical exponents

τβ ∝ |σ|−1/2a (2.12)

τr ∝ |σ|−γ , γ =1

2a+

1

2b(2.13)

The second equation has been presented already in connection with the temperature de-

pendence of structural relaxation. The fit of the temperature dependence of the viscosity,

which is proportional to τr, by a power law is shown in Fig. 2.7. The important re-

sult of the MCT is that both the temperature dependence of the relaxation times and

the time dependence of the relaxation function are determined by two system-dependent

parameters: λ and Tc.

2.4. Description of the LGT by the mode-coupling theory 15

Figure 2.7: Arrhenius plot of viscosity η in Poise of the mixed salt 0.4Ca(NO3)2 ·0.6KNO3.

The dashed line is a fit by an Arrhenius law. The full line represent a power law (see

text) with γ = 4.06. Experimental data was obtained by [WBM69] and fit was done by

[GS92].

A weak point of the basic MCT is that it predicts a complete arrest of α relaxation for

temperatures below Tc which is not observed in experiments. Extended versions of MCT

were elaborated to avoid these difficulties. In this theory the particle can participate,

together with kinetic motion, in thermal hopping diffusion which results in avoiding the

critical behavior at Tc. The ratio of hopping and kinetic mobility defines the behavior

of the supercooled liquid at the region around Tc. Below Tc only thermally activated

diffusion has to be taken into account. Then the temperature dependence of the charac-

teristic relaxation times can be qualitatively described by an Arrhenius law for T < Tc(see

Fig. 2.7). The hopping mobility defined by the energy barrier could be different for dif-

ferent types of motion, resulting in the absence of a universal time scale for the region

below Tc.


Figure 2.8: Temperature dependence of the various relaxation times for o-terphenyl. α

relaxation: dielectric relaxation (♦), nuclear magnetic resonance (•); slow β relaxation:

dielectric relaxation (¨); fast β relaxation: neutron scattering (¥). Lines are guides for

the eye. Data were taken from [EAN96].

2.5 Slow β relaxation

In addition to the structural relaxation, another relaxation process was found at tem-

peratures near and below Tg. This process, slow β relaxation, was observed for many

glass formers including polymers and organic low-molecular weight compounds. The β

process is observed usually in the kHz frequency range. Its temperature dependence is

well described by an Arrhenius law. The temperature dependence of the relaxation rate

of the slow β relaxation is shown in Fig. 2.8 for the typical glass former o-terphenyl.

The origin of slow β relaxation is not yet understood. Several authors suggest in-

tramolecular motion as a source of the β process [Wu91, EAN96, ARCF96]. This ex-

planation works well in the case of polymers. However, it can not explain the β process

in glass formers consisting of rigid molecules. The systematic study of simple molecular

glass formers by Johari and Goldstein [JG70, Joh73] let them conclude that the slow β

relaxation is an intrinsic property of glasses. They considered it to be due to localized

motions in the regions of loose packing of molecules in the disordered structure. This

model assumes that slow β relaxation is a precursor of structural α relaxation. This idea

2.5. Slow β relaxation 17

was used in MCT together with the name “β relaxation”. However, the experimental

study shows that the fast β relaxation which is observed in the GHz frequency range

and slow β relaxation are different processes and the explanation which was suggested

by Johari and Goldstein corresponds to the fast β relaxation. The understanding of the

molecular nature of slow β relaxation is therefore still an open question.

The experimental study of the slow β relaxation is difficult due to a lack of proper

experimental techniques to access various types of molecular dynamics with microscopic

space resolution and sensitivity to slow dynamics. The most commonly used dielectric

spectroscopy and NMR follow mainly the reorientation of molecules and do not provide

an atomic-scale space resolution. Complementary information on translational relaxation

on a molecular length scale can be obtained with neutron scattering. However, it hardly

can access the time region above ∼1 ns, therefore only few measurements have been

reported [ARCF96].

Chapter 3

Nuclear forward scattering (NFS) of

synchrotron radiation (SR)

The purpose of this chapter is to introduce the method of nuclear forward scatter-

ing (NFS). At first the Mossbauer effect, which is the basis of nuclear resonant scattering,

is described. Then the principles of NFS are introduced. NFS in the static case, when

the evolution of the nuclear ensemble is governed by a time independent Hamiltonian, is

described in the third part. The next section is devoted to the influence of spatial dynam-

ics on NFS. In particular, the influence of Kohlrausch relaxation has been developed. In

the last part the application of NFS to study dynamics in non-resonant samples is briefly

described.

3.1 The Mossbauer effect

The Mossbauer effect is the recoil-free emission or absorption of nuclear gamma radiation.

It was discovered by Rudolf L. Mossbauer in 1958 [Mos58]. A γ-quantum emitted in the

de-excitation of a nucleus bound in a solid can be absorbed by another nucleus of the

same kind and excite it in a resonant way.

The classical point of view implies that the emission of a γ-quantum is connected with

a recoil of the nucleus in order to conserve momentum. If E0 is the energy of the excited

state then the energy of the γ-quantum is Eγ = E0 − ER. From energy and momentum

conservation the recoil energy ER of a free nucleus can be calculated as

ER =E2

γ

2Mc2(3.1)

where M is the mass of the nucleus. Similarly, when a γ-quantum is absorbed it loses the

energy ER due to recoil of the nucleus. In order to find whether resonant absorption will

occur in spite of the losses due to recoil, the energy gap 2ER has to be compared to the

19

20 Chapter 3. Nuclear forward scattering (NFS) of synchrotron radiation (SR)

sharpness Γ0 of the absorption (emission) process. This value is related to the life time

τ0 of the excited state by Heisenberg’s uncertainty principle:

Γ0 =~

τ0(3.2)

Typical values for the life time are 101−103 ns (τ0 = 141 ns for the 57Fe), and accordingly

Γ0 is of the order of 10−7 − 10−9 eV. The recoil energy depends strongly on the energy of

the γ-quantum, which is rather large (Eγ = 14.4 · 103 eV for the 57Fe). Then the energy

shift 2ER is of the order of 10−3 − 10−2 eV, which is five orders of magnitude larger than

the line width Γ0. Consequently, the resonance condition can not be fulfilled for free

nuclei.

The situation is different when emitter and absorber nuclei are bound in a solid. As

the binding energy EB of atoms is of the order of eV, the nuclei cannot recoil freely

(ER ¿ EB), but only with creation or annihilation of phonons. The quantum mechanical

concept implies that there is a probability for the nuclei to emit (absorb) γ-quanta without

phonon excitations and, as a consequence, without change of energy. The energy of

the γ-quantum is Eγ = E0, the same for emission and absorption, and the resonance

process can occur. The probability of the emission without recoil is called the recoilless

fraction or Lamb-Mossbauer factor fLM . Recoilless (zero-phonon) emission, absorption,

and scattering of nuclear γ radiation is used in Mossbauer spectroscopy (MS).

3.2 Principles of NFS

The main line of development of MS was absorption spectroscopy. Here the ensemble

of nuclei manifests itself as a set of independent scatterers. Therefore no information

about the collective state of the nuclei (relative phases of the nuclei in the ensemble)

can be obtained. Nevertheless, the unique small ratio of line width of the excited level

Γ0 to the transition energy E0 attracted attention to the possibility of realizing coherent

effects in the ensemble of Mossbauer nuclei. It demands another type of experiment -

the scattering of γ-quanta by an ensemble of nuclei. The theory of nuclear resonant scat-

tering (NRS) was introduced and developed mainly by Kagan, Afanas’ev and Kohn and

Trammell and Hannon in the 60’s and 70’s; for recent reviews see [Kag99, HT99]. The

first attempts to observe coherent phenomena in an ensemble of nuclei were done with

traditional Mossbauer sources; for a review see [Smi86]. After synchrotron radiation (SR)

was introduced as a new tool for scientific research, it was suggested by Ruby [Rub74]

to use it for exciting Mossbauer nuclei. The first successful nuclear resonant scattering

experiment using SR was performed by Gerdau et al. [GRW+85] in 1985. This experi-

ment was done in Bragg reflection geometry from a single crystal of yttrium iron garnet.

3.2. Principles of NFS 21

During the next 6 years investigations of resonant scattering in Bragg and Laue geome-

try attracted worldwide interest. These measurements, however, required single crystals

of high quality, enriched in the Mossbauer isotope (e.g. 57Fe), and exhibiting pure nu-

clear Bragg reflections. These restrictions on the sample material were removed by using

nuclear forward scattering (NFS). NFS was observed for the first time by Hastings et

al. [HSvB+91] in 1991. NFS opens the way to study a wide range of materials. In

particular relaxation and diffusion were investigated in crystalline, biological and glassy

samples [MFW+97, TW99, LW99, VS99, ASF+01].

Nuclear resonant scattering of SR by a single nucleus is a process which can be sepa-

rated into two stages. Due to the short duration of the SR pulse compared to the excited

state lifetime τ0, the pulse creates an excited nuclear state. The second stage is the decay

of this state with re-emission of radiation. The time evolution of the observed scattered

radiation is described by a natural decay, i.e., an exponential decay with the lifetime τ0.

Time-independent hyperfine interactions (electric quadrupole interaction and magnetic

dipole interaction) between the nucleus and the surrounding electrons split the nuclear

states into several sub-levels, which lead to indistinguishable scattering paths. This re-

sults in a modulation of the exponential decay of the scattered intensity by a temporal

interference pattern called quantum beat (QB).

The scattering by an ensemble of nuclei becomes more complicated. An incoming

photon can now interact with any nucleus in the ensemble. In the case of elastic scattering

of a photon, where the intrinsic state of the scattering system stays unchanged, it is

impossible to ascertain which nucleus in the ensemble was excited. As a consequence, a

collective state of the ensemble is involved in the scattering process. The collective state

formed after absorption of the photon is called nuclear exciton. The nuclear ensemble

behaves like a macroscopic resonator whose properties differ qualitatively from those of

the individual nuclei. This shows up by changes of the time and space distribution of the

scattered intensity.

When the excitation is distributed over the entire nuclear ensemble, interference be-

tween wavelets re-radiated by the nuclei occurs and a coherent radiation field is built up.

The scattering of the photon by the entire nuclear ensemble is the spatially coherent chan-

nel of scattering, whereas the incoherent channel is formed by scattering on the individual

nuclei. Due to the usual property of the coherent scattering, the intensity of scattering is

proportional to the square of the number of nuclei. Another property is that an angular

interference pattern of scattering appears. Fully constructive interference occurs in the

case of an ordered ensemble of nuclei in all directions where the wavelets are in phase. In

the case of the disordered ensemble of nuclei, like glasses and liquids, which are subject

of interest in this work, constructive interference appears only in the forward direction of

scattering. The spatially coherent nuclear resonant scattering in the forward direction is


called nuclear forward scattering.

3.3 Theory of NFS in the static case

Usually the amplitude (intensity) of NFS is calculated in the frequency domain [KAK79].

Here the scattering amplitude is defined as a function of energy. After that, using the

δ-function shape of the SR pulse, the Fourier transformation of the scattering amplitude

can be performed resulting in the time-dependent amplitude of the scattered radiation.

However, in order to emphasize the dynamical aspects of NFS, we use a method which

allows to calculate the amplitude straightforward in the time domain. This method has

been developed by Shvyd’ko [Shv99a, Shv99b]. The intensity equals to the square of the

scattered amplitude which can be written as a sum (see eq. (2.34) in [Shv99b])

E(t) =∞∑

k=1

(−1)k ξk

k!E(k)(t) (3.3)

where ξ = σ0N0L/4 is named the effective thickness in this work. It is proportional to the

resonance cross section σ0, the number of the resonant nuclei per unit volume N0, and the

real thickness L of the target. Each term E(k)(t) corresponds to a channel of scattering

with k collision events between the photon and the nuclear ensemble. This term is given

by the recursion relation (see eq. (2.36) in [Shv99b])

E(1)(t) =E0

τ0K(t, 0) (3.4)

E(k+1)(t) =1

τ0

∫ t

0

dt · E(k)(t) ·K(t, t) (3.5)

where E0 =√

I0/∆ω is the amplitude of the electromagnetic wave coming to the target

from the SR source within the frequency band ∆ω determined by the monochromator, I0

is the corresponding intensity and K(t, t) is the self-correlation function which takes into

account the change of spin and spatial states of the nuclear ensemble in the scattering

event. In this work we assume that the time evolution of the nuclei is governed by

time-independent or stochastic forces. Then, the exact dependence of the self-correlation

function on t and t reduces to the dependence on their difference K(t, t) → K(t− t).

3.3.1 Single resonance

In this section we assume that the nuclear ensemble is in the static state, i.e., the spatial

position of each nucleus is fixed and does not change with time. If we also assume the

absence of hyperfine interactions (single resonance case) then the self-correlation function

3.3. Theory of NFS in the static case 23

reduces to (see eq. (2.42) in [Shv99b])

K(t) = e−t/2τ0 (3.6)

that reflects the decay of the excited state. Inserting this expression for the self-correlation

function into eqs. (3.4) and (3.5) gives the possibility to obtain the amplitude of the

scattered radiation. At first, we consider the single scattering (kinematical) approximation

of the amplitude which is very convenient for the analysis of physical problems. This

approximation is obtained by retaining only the first term in the general solution (3.3).

The intensity, which is the square of the amplitude, is given as

I(t) =I0∆ω

ξ2

τ 20

e−t/τ0 (3.7)

The time evolution of the delayed intensity is described by a natural decay with the

lifetime τ0.

Also an exact analytical solution of eq. (3.3) which takes into account multiple scat-

tering of the photon by the nuclear ensemble, can be obtained for the single resonance.

The integration in eq. (3.5) can be done analytically and one obtains for the amplitude

the well known result [KAK79]

E(t) = E0ξ

τ0e−t/2τ0σ(ξt/τ0) (3.8)

σ(x) =J1(2

√x)√

x(3.9)

where J1(x) is the Bessel function of first kind and first order. This amplitude oscillates

around zero with time while decaying exponentially. The zero points of this oscillations

corresponding to ξt/τ0 = 3.7, 12.3, . . . are defined by the roots of the Bessel function.

Such a beat is called dynamical beat (DB). The time dependence of the NFS intensity

can then be obtained as a square of the amplitude:

I(t) =I0∆ω

e−µeL ξ2

τ 20

e−t/τ0σ2(ξt/τ0) (3.10)

where the decrease of intensity due to the electronic absorption in the sample with the

absorptions length µe is taken into account. Some characteristic features of the DB

are [vB99]: the DB is aperiodic, the apparent periods increase with time; the DB periods

decrease with increasing effective thickness; the initial decay is sped up proportionally to

the effective thickness.

If the value ξt/τ0 is relatively small ( small effective thickness of the sample or short

experimental time window) then a solution is obtained, which is called quasi-kinematical

approximation in this work. The function σ(ξt/τ0) can be approximated by an exponent

σ(ξt/τ0) ≈ exp(−ξt/2τ0) so that the intensity is replaced by [vBSH+92]

I(t) = I(0)e−(1+ξ)t/τ0 (3.11)


Figure 3.1: Time evolution of the NFS intensity (solid line), its kinematical (dotted line)

and quasi-kinematical (dashed line) approximation. The state of the nucleus is assumed

to be unsplit and the effective thickness is ξ = 5.

This approximation is valid for ξt/τ0 . 0.75. One can estimate the characteristic decay

time as

τd =τ0

1 + ξ(3.12)

so that with increasing effective thickness the characteristic decay time decreases. This

phenomenon, called speed-up effect, is a property of the coherent superposition of multiple

scattering processes. The time dependence of the NFS intensity in full form, kinematical

and quasi-kinematical approximation are shown in the Fig. 3.1. One easily sees that the

full NFS intensity decays faster than the one in the kinematical approximation even at

small times. One can conclude that for ξ ∼ 1 the kinematical approximation is unrealistic

and one has to use the quasi-kinematical approximation.

3.3.2 Hyperfine splitting

Time-independent hyperfine interactions lead to an additional factor in K(t). We limit

our consideration to the quadrupole interaction defined by an axially symmetric electric

field gradient (EFG). This type of the interaction is described in Appendix B. The

quadrupole interaction splits the excited state of the 57Fe nucleus into two sub-levels. Also

we assume that the EFG is isotropically distributed over the nuclear ensemble. Then the

3.3. Theory of NFS in the static case 25

self-correlation function K(t) can be written as

K(t) = e−t/2τ0 · cos (Ωt/2) (3.13)

and the intensity in the kinematical approximation is given by

I(t) =I0∆ω

ξ2

4τ 20

e−t/τ0(1 + cos(Ωt))/2. (3.14)

The time evolution of the intensity is described by a natural decay modified by a quantum

beat (QB) which is periodic in time. The effective thickness is decreased by a factor of

two compared with the single resonance case (see eq. (3.7)). Phenomenologically one can

explain such a behavior by separating all nuclei into two equal fractions with the resonant

energies E0+~Ω/2 and E0−~Ω/2, respectively. Two indistinguishable scattering channels

appear and scattering of a photon through each of them results in a wavelet of a single

energy. The interference of the wavelets leads to a QB.

By contrast to the single resonance case, the general solution for the amplitude (see

eq. (3.3)) in presence of hyperfine interactions can not be obtained analytically. Numer-

ical calculations based on the iterative procedure given by eqs. (3.4),(3.5) have to be

performed. However, there is an analytical approximation which is valid for small values

of ξ/Ωτ0. This approximation is important for the experiments presented in this work.

The parameter of approximation can be explained as following. The absorption/emission

probability of each sub-level is described in the energy domain by a Lorentzian with line

width Γ0. This leads to the overlap of the probabilities of two sub-levels separated by

~Ω. The area of the overlap region is proportional to Γ0/~Ω = 1/Ωτ0. This overlap gives

the possibility for a photon scattered by one sub-level be re-scattered by another one.

This process called cross-scattering becomes stronger with increase of multiple scattering

which is proportional to the effective thickness ξ. Combining the two proportionalities

we obtain the parameter ξ/Ωτ0. In the first order of the parameter the intensity can be

approximated as

I(t) ' I0∆ω

e−µeL · ξ2

4τ 20

· e−t/τ0 σ2(ξt/2τ0) · (1 + cos(Ωt+ ξ/2Ωτ0))/2 (3.15)

The expression for the intensity factorizes into a term describing the single resonance

intensity (see eq. (3.10)) with half effective thickness and into a QB term with additional

phase shift. This shift of the QB was confirmed in experiments in Laue geometry diffrac-

tion [CSZ+92] and in NFS [vBSH+92]. The validity of this approximation is checked in

Fig. 3.2. Here the numerical simulation of the intensity according to eqs. (3.3)-(3.5) is

presented for various values of ξ/Ωτ0. One can see that the minimum positions of the

QB are shifted to smaller times with increasing of the thickness. Additionally, the single-

resonance intensity with the effective thickness ξ/2 is shown. This intensity, according to


Figure 3.2: Time evolution of the NFS intensity. The solid line represents the numerical

simulation of the intensity for the sample with quadrupole splitting and effective thickness

ξ. The thick solid line represents the single resonance intensity (see eq. (3.10)) with

effective thickness ξ/2. The parameter of the quadrupole splitting ~Ω is chosen as 24Γ0.

The effective thickness is ξ = 0.5 (a), 5 (b), 10 (c) and 20 (d). The vertical dotted lines

show the positions of the QB minima for (a).

eq. (3.15), has to be an envelope of the numerical simulation. It is seen that this is valid for

ξ/Ωτ0 = 0.02, 0.2, 0.4. However, when ξ/Ωτ0 becomes comparable to 1 (ξ/Ωτ0 = 0.83 in

the figure) the approximation fails. In this case the interaction of QB and DB structures

changes strongly the shape of the intensity.

3.4 Influence of spatial dynamics on NFS

Two important features of NFS are exploited in order to obtain information about con-

densed matter dynamics. NFS is a nuclear spectroscopy technique, like nuclear magnetic

resonance (NMR), nuclear quadrupole resonance (NQR) and MS. Being such NFS ob-

serves the time evolution of the hyperfine field at the nucleus via the time evolution of

3.4. Influence of spatial dynamics on NFS 27

the nuclear spin. The relaxation processes which involve the hyperfine field are reflected

in the spin relaxation and, consequently, in the shape of the scattered intensity. On the

other hand, the very small width of the resonance makes NFS as well as MS sensitive

to the atomic motion in the time scale up to µs - diffusive motion. This sensitivity to

the time dependence of the atom’s position makes NFS similar to quasi-elastic neutron

scattering.

In MS these two dynamical processes are often considered separately. However, it was

pointed out in [Dat75], that in the case of liquid dynamics it is important to take into

account both spin relaxation and diffusive motion. In this work we will call the diffusive

motion spatial relaxation in order to emphasize that the molecular motion in a glass

forming liquid does not obey the classical diffusion law.

3.4.1 General formalism

Using the density matrix formalism one can write the self-correlation function K(t)

as [Dat75]

K(t) = e−t/2τ0 · Tr(ρA(0)eikR(0)A+(t)e−ikR(t)) (3.16)

where A(t) is an operator which acts only on the nuclear spin state at time t and defines the

intensity and the polarization of the emitted or absorbed radiation, R(t) is the coordinate

of the center of mass of the nucleus at time t and ρ is the equilibrium density matrix for

the entire system which can be factorized into the density matrix of the nuclear spin states

ρn and the density matrix of the surroundings ρs. The self-correlation function can be

influenced by two different relaxation mechanisms: relaxation in the spin space is seen

by the operator A(t) and spatial relaxation is seen via the position of the nucleus R(t).

If the nuclear spin dynamics is completely independent of the spatial dynamics, the spin

and spatial operators commutate in eq. (3.16) and K(t) factorizes into spatial and spin

parts

K(t) = e−t/2τ0 · Tr(ρnA(0)A+(t)) · Tr(ρseikR(0)e−ikR(t)))

= e−t/2τ0 ·G(t) · Fs(k, t)(3.17)

Here G(t) represent the self-correlation of the nuclear spin in time and can be called

perturbation factor in analogy to Time Differential Perturbed Angular Correlation as will

be seen in the next chapter. Fs(k, t) is called van Hove self-intermediate function [vH54]

and represents the evolution of the spatial position of the nucleus in time according to

the theory of Singwi and Sjolander [SS60].

The absolute value of the spin is given by the nuclear state, its direction is defined by

the experienced hyperfine fields. If no field is seen by the nucleus, the spin direction and

its time evolution are undefined. In this case the perturbation factor G(t) equals 1. When


a time-independent hyperfine field acts on the nucleus, the spin precesses around the field

direction, resulting in a periodic oscillation of the perturbation factor. For quadrupole

interaction, which is considered in this work, G(t) = cos(Ωt/2). This information was

already used in the Section 3.3.2. Relaxation of hyperfine fields results in spin relaxation,

which is expressed by the perturbation factor. Theoretical investigation of spin relaxation

applied to Mossbauer spectroscopy have been performed e.g. in [AK64, Blu68, Dat81]. A

recent study of spin relaxation in NFS can be found in [LW99].

The factorization in eq. (3.17) is only allowed if spin and spatial time evolution are

independent. Otherwise the general expression (3.16) for K(t) has to be considered.

An example of the coupling between spin and spatial relaxation will be considered in

the following. In molecules the direction of the electric field gradient on the nucleus is

defined by the specific molecular structure. Therefore, any rotational molecular relaxation

strongly couples with the spin. If the process which governs rotational motion also leads

to the change of the nuclear position R(t) then this process is seen by both spin and

spatial operators in K(t) and the general expression has to be applied. The theoretical

study of this process will be given in the Chapter 4.

3.4.2 Debye relaxation

In this section we neglect the spin part of the self-correlation function K(t), assuming the

absence of hyperfine interactions. Then the self-intermediate function Fs(k, t) defines the

time-evolution of the scattered intensity. It can be factorized into parts corresponding to

fast and slow nuclear motion relative to the experimental time window. The region of the

observation time is limited from the short-time side by 1÷10 ns for technical reasons and

from the long-time side by a few µs due to the natural decay. Fast motions that happen

on short (picosecond) time scales cannot be detected directly. Instead they give rise to

the Lamb-Mossbauer factor fLM with

Fs(k, t) = fLM Fs(k, t) (3.18)

where Fs(k, t) describes the slow motion of the nuclei that can be observed directly in the

experimental time window. A similar factorization is used in neutron scattering where

fast motions are described by the Debye-Waller factor fQ and the momentum transfer Q

is used instead of the full momentum of the scattered photon.

The absolute value of the wave vector k in eq. (3.18) is fixed by the energy of the excited

state (k ≈ 7.3 A−1 for 57Fe). Therefore we omit the dependence of the self-intermediate

function on k and replace Fs(k, t) by Fs(t).

The factorization in eq. (3.18) goes through the recursion relation (3.5) and results

in replacing of the effective thickness ξ by ξfLM . Qualitatively one can say that only a


Figure 3.3: a)Distribution of Debye relaxators corresponding to a Kohlrausch relaxation

function for λt = 1/τ0 and several β; b) distribution of Debye relaxators corresponding

to a Kohlrausch relaxation function scaled by λt and area to the same position of the

maximum point for different β.

fraction fLM of all nuclei in the sample takes part in the scattering process.

The influence of slow motion to NFS has been theoretically investigated by Smirnov

and Kohn [SK95, KS98, KS99] for several types of Fs(t): free diffusion governed by Debye

(exponential) relaxation, diffusion restricted in space, and diffusion governed by a set of

Debye relaxators.

The most simple type of the intermediate function is Debye relaxation Fs(t) = exp(−λtt)

with a characteristic relaxation rate λt = Dk2 where D is the diffusion coefficient. In this

case the time evolution of the intensity reduces to the analytical expression [KS99]

I(t) = I(0)e−t/τ0 · e−2λttσ2(ξfLM t/τ0) (3.19)

where I(0), the intensity at zero time, is the same as in the static case (see eq. (3.10)) mul-

tiplied by f 2LM . One can see that for Debye relaxation the intensity factorizes into the re-

laxation and multiple scattering. The DB, which is a pronounced feature of the NFS inten-

sity, is not influenced by relaxation. When the effective thickness or the Lamb-Mossbauer


Figure 3.4: Simulation of the NFS intensity in the kinematical approximation. The

relaxation is governed by the Kohlrausch relaxation function. The parameters of this

function were chosen to have the same position of the main peak of the distribution of

Debye relaxators for different β, similar to Fig. 3.3b.

factor of the sample is small, the expression (3.19) reduces to the quasi-kinematical ap-

proximation

I(t) = I(0) e−(1+2λtτ0+ξfLM )t/τ0 (3.20)

The intensity decays faster than the natural decay. The additional decay is formed by the

superposition of multiple scattering and relaxation. It is not possible to separate the terms

associated with these two processes in the frame of quasi-kinematical approximation.

3.4.3 Kohlrausch relaxation

The description of the liquid-to-glass transition requires another type of Fs(t), a Kohlrausch

relaxation function (see eq. (2.6))

Fs(t) = e−(tλt)β

(3.21)

which describes the final decay of the relaxation function in amorphous materials. The

stretching parameter β varies in the region 0 < β 6 1. For this type of relaxation we have

developed an special approach [SFA+02] which is presented below.

The NFS intensity in the kinematical approximation can be written as

I(t) = I(0)e−t/τ0F 2s (t) = I(0)e−t/τ0 · e−2(λtt)β

(3.22)


Figure 3.5: Simulation of the NFS intensity normalized to the natural decay. The relax-

ation is governed by Kohlrausch function with β = 0.5. The effective thickness parameter

was chosen as ξfLM/τ0 = 1.

The Kohlrausch relaxation function can be written as a distribution of Debye relax-

ation processes [GS92]

Fs(t) =

∫ ∞

0

dλ e−λtρ(λ) (3.23)

where the distribution ρ(λ) corresponds to the probability to find a process with the

relaxation rate λ. The distribution ρ(λ) is shown in Fig. 3.3a for different values of β and

λt = 1/τ0. The Debye law (β = 1) corresponds to the point distribution

ρ(λ) = δ(λ− λt). (3.24)

For β < 1, the shape of the distribution is a non-symmetrical peak with a long high-

frequency tail defined by the von Schweidler law ρ(λ) ∝ (λ/λt)−1−β and a cut-off at the

low frequency. The position of this peak shifts to lower frequencies when β decreases.

To compare the influence of β, the distribution, scaled to have the same amplitude and

position of the maximum for different β, is shown in the Fig. 3.3b. One can see that

the high-frequency tail increases strongly with decreasing β and there is a slight increase

of the low-frequency tail. Such behavior of the distribution function defines the shape

of the intensity in the kinematical approximation (Fig. 3.4). The Debye relaxators with

different frequencies play a role in different time regions. Close to time zero, the shape

of the intensity is defined by the high-frequency part of the distribution, whereas the

low-frequency part corresponds to large times. In Fig. 3.4 the values of λt were chosen as

in Fig. 3.3b with position of the distribution maximum at λ = 1/τ0. This fact results in


Figure 3.6: Relative shift of the DB structure minima as a function of λt for β = 0.5. In

the inset the same curves are shown in a log-log plot. Additionally, the relative shift for

the first DB structure minimum is shown for β = 0.3 and 0.7.

roughly the same slope of the intensity at the region of time around t = τ0 for different

β. Before this time, the high frequency tail dominates the intensity and there is a strong

decay for small values of β compared to β = 1. After t = τ0, only a weak change of

the exponential slope is observed due to roughly the same position of the cut-off of the

distribution in the low-frequency region. Qualitatively, the same result was obtained

in [KS98] for the case of a discrete distribution of Debye relaxators which describe jump

diffusion in a crystal.

The influence of Kohlrausch relaxation on the time evolution of the intensity in the

general case is more complicated. It is different from the influence of the Debye relaxation

where the multiple scattering factor enters multiplicatively. The numerical simulation of

the intensity as a function of time is shown in Fig. 3.5 for β = 0.5 and different λt. As

opposed to the simple exponential relaxation, the relaxation governed by the distribution

of the exponential relaxators leads to a shift of the DB structure to later times. It is

clearly seen from Fig. 3.5 for the Kohlrausch relaxation function. Also it has been shown

in [SK95, KS98] for the diffusion described by a discrete set of the exponential relaxators.

The analysis of the influence of the relaxation governed by the Kohlrausch function to

the multiple scattering factor of the delayed intensity can be done in the approximation

of small values of the relaxation rate λt, when the Kohlrausch relaxation function reduces

to the von Schweidler function. It is shown in Appendix A that the time evolution of the


X - r a y sF o i l 1

F o i l 2S a m p l e

D e t e c t o rq

Figure 3.7: Experimental setup of TDI. Two single line resonant foils are placed before

and after the sample with the first mounted on a Mossbauer drive in constant velocity

mode. Detector measures the radiation scattered in angle ϑ and corresponded momentum

transfer Q.

delayed intensity can be written as

I(t) = I(0)e−t/τ0 ·(

σ(ξfLM t/τ0) − (λtt)βσβ(ξfLM t/τ0)

)2(3.25)

where σx(z) is a generalized version of the function σ(z) ≡ σ1(z). It is introduced in

eq. (A.20). This expression gives the possibility to obtain the shift of the positions of the

DB minima as a function of the relaxation rate. If ti is the time corresponding to the ith

minimum of the delayed intensity for the static case and ∆ti is a shift of this minimum

from its static position due to the relaxation then

∆titi

= (λtti)βCi(β) (3.26)

Ci(β) =σβ(ξfLM ti/τ0)

ξfLM ti/τ0 · σ′(ξfLM ti/τ0)(3.27)

The relative shift of the minimum position is proportional to λβt with a proportionality

coefficient that depends only on β. For the extremal case of β = 1 (Debye relaxation),

the coefficient Ci(1) = 0 which corresponds to zero shift. The numerical simulation of

the relative shift of the DB structure minima positions for the relaxation governed by

Kohlrausch function is shown in Fig. 3.6. The inset of this figure shows the log-log plot

of the region near λt = 0. It shows that at the region of small λt, the relative shifts obey

eq. (3.26). When λt becomes comparable with ξfLM/τ0, the relative shifts begin to diverge

to large times. The root-like dependence of the relative shift for small λt means that an

infinitely small value of the relaxation rate results in a certain (not infinitely small) value

of the relative shift. It gives the possibility to extract information about relaxation from

multiple scattering corrections of the NFS intensity in the temperature region where the

relaxation is very small.


Figure 3.8: Scattering from glycerol measured in the structure factor maximum Q =

1.5 A−1 at different temperatures [BFM+97].

3.5 Time domain interferometry

At this place we would like to mention briefly that nuclear forward scattering can even be

used to determine spatial relaxation in non-resonant samples, using Time Domain Inter-

ferometry (TDI). This interferometric method is the time analog of Rayleigh Scattering

of Mossbauer Radiation (RSMR); for a review on RSMR see e.g. [Cha79, KGNP90].

The method was developed by Baron et al. [BFM+97] and theoretically described by

Smirnov et al. [SKP01]. Fig. 3.7 shows the experimental setup of TDI. A synchrotron

pulse excites a resonant foil 1, then is scattered by the sample with the momentum trans-

fer Q, depending on angle ϑ, and excites the resonant foil 2. The resonant energies of

the two single-resonance foils are shifted relative to each other by ~Ω using a Mossbauer

drive operated in constant velocity mode. After excitation by a synchrotron pulse, nu-

clear scattering will show a temporal interference pattern equivalent to a QB. The sample

placed between the foils stochastically perturbs the phase shift between the wavelets scat-

tered by the two foils. This leads to a damping of the QB, which is described by the

self-intermediate correlation function S(Q, t) of the sample. For two identical foils and

~Ω larger than the resonant line width the scattering intensity can be written as [FPB99]

I(Q, t) ∝ |R(t)|2 (S(Q) + S(Q, t) cos Ωt) (3.28)

3.5. Time domain interferometry 35

where S(Q) is the static structure factor of the sample and |R(t)|2 is the intensity scattered

by one resonant foil.

The results of TDI measurements on the glass forming liquid glycerol are shown in

Fig. 3.8. The envelope of the QB structure is the self-intermediate function, which exhibits

a stretched exponential decay. As temperature increases the characteristic relaxation rate

increases and the QB becomes more strongly damped.

This technique can be used to study quasi-elastic scattering on time scales of ∼5 ns to

∼500 ns (energy scales of ∼150 neV to ∼1.5 neV) with large momentum transfer (up to

14 A−1). The inherent brilliance of SR permits one to do experiments with small (∼1 mm)

samples and with extreme (µrad) collimation, which is not possible for RSMR.

Chapter 4

SR-based perturbed angular

correlation (SRPAC)

This chapter is dedicated to the description of a new method in the family of NRS methods

- SR-based Perturbed Angular Correlation (SRPAC). This method is based on incoherent

NRS, which is described in the first section. The principles and theoretical description of

SRPAC are discussed in the two following sections. The theory to describe the influence of

rotational relaxation on SRPAC is developed in the next chapter. The last part explains

how the same theoretical approach can be applied to explain the influence of rotational

relaxation on NFS.

4.1 Spatially coherent versus incoherent nuclear res-

onant scattering

Incoherent NRS was studied extensively during the last 10 years in connection with

the possibility to investigate lattice vibrations. Pioneering work on this method, now

called nuclear inelastic scattering (NIS) were done by Seto et al. [SYK+95], Sturhahn et

al. [STA+95] and Chumakov et al. [CRG+95]. This method uses incoherently scattered

intensity integrated in time.

The time evolution of incoherent NRS of SR has been investigated less thoroughly.

Theoretically the incoherent channel of NRS was studied by Trammell and Hannon [TH78],

Iolin [Iol96] and Sturhahn and Kohn [SK99]. The first experimental observations of the

time evolution of incoherent scattering were done by Bergmann et al. [BHS94] and by

Baron et al. [BCR+96].

There are several possible channels of incoherent nuclear resonant scattering: nuclear

resonant fluorescence, conversion electron emission and atomic fluorescence following nu-

clear absorption [SK99]. They are shown in Fig. 4.1. The difference between these chan-

37

38 Chapter 4. SR-based perturbed angular correlation (SRPAC)

γ

γ

Xe-

I

III IV V

II

Figure 4.1: Incoherent NRS. Steps: I - resonant absorption of the γ-photon with Eγ '14.4 keV for 57Fe, II - reemission of the γ-photon with Eγ ' 14.4 keV (nuclear resonance

fluorescence), III - transfer of the excitation energy to the electron shell, IV - emission

of the electron and formation of a hole, V - filling of the hole by another electron and

emission of an X-ray photon.

nels is in the type of radiation following the de-excitation of the nucleus.

Nuclear resonant fluorescence corresponds to the reemission of the photon by the

nucleus (II in the figure). Another channel of de-excitation, the internal conversion,

is the transfer of the excitation energy to the electron shell (III). After a conversion

electron has been emitted (IV), the remaining ion core will de-excite by emission of X-

rays (V). The conversion coefficient α gives the ratio of internal conversion to resonance

fluorescence. For 57Fe α = 8.2, and the conversion channel of the scattering is much

stronger than nuclear resonant fluorescence. However, due to the complexity of the process

of atomic fluorescence, X-rays lose information about the nuclear process. The conversion

electrons undergo several collisions with large deflection angles before emanating from

the surface and have to be averaged over all directions of emission which leads also to a

loss of information about the nuclear scattering [SQTA96]. Therefore, in this study we

concentrate on the nuclear resonant fluorescence and do not consider conversion channels

at all. In the following the term incoherent scattering of SR will mean only nuclear

resonant fluorescence.

Incoherent NRS is characterized by the possibility to identify the nucleus which scat-

ters the photon. The necessary condition for it is that the nucleus must change its state

in the scattering process and is therefore tagged. This can happen by a change of spin

of the ground state (spin-flip process) or by a change of the vibrational state of the

nucleus. Coherent scattering, by contrast, is de-localized over the whole ensemble of nu-

clei [TH78, KAK79]. There is no change of state of any nucleus due to scattering. An

exception could be coherent inelastic scattering which has not been observed up to now

4.1. Spatially coherent versus incoherent nuclear resonant scattering 39

due to the short lifetime of phonons compared with the nuclear lifetime [CBR+98].

NFS is elastic scattering and is proportional to the square of the Lamb-Mossbauer

factor fLM2. Incoherent scattering can happen with absorption/emission of phonons.

This fact gives the possibility to measure the phonon density of states [CRG+95, STA+95]

by tuning the energy of the incoming beam with respect to the nuclear resonance energy.

If the energy width of the incoming beam is large enough to cover all phonon assisted

processes, then incoherent scattering does not depend on fLM at all.

The delocalization of coherent scattering results in a strong collimation of the scattered

beam. If the target is formed by an irregular ensemble of nuclei, then scattering exists only

in the forward direction. Incoherent scattering, localized on a certain nucleus, can happen

in any direction. The probability of scattering under a certain angle is defined by the spins

of the ground and excited states of the nucleus and the direction, the multipolarity and

the polarization of the incoming photon.

Incoherent scattering is single nucleus scattering. If no hyperfine interactions exist then

the time evolution of the intensity is an exponential decay with a characteristic lifetime

τ0, i.e. a natural decay. The cross-section of the resonant absorption of the photon by a

single nucleus is relatively small, and one can usually neglect multiple scattering.

Another situation appears in coherent scattering. The cross-section of the absorption

of a photon by the collective ensemble of nuclei is relatively high, which results in an

essential contribution of multiple scattering. It changes the time evolution of scattering

which exhibits a speed-up effect and dynamical beats (see preceding chapter on NFS).

When hyperfine interactions split ground and excited states of the nucleus, experimen-

tally indistinguishable paths of scattering appear which interfere. Therefore the evolution

of the scattered intensity is modulated by QB. The interference pattern is different for the

coherent and incoherent channels. Incoherent scattering is sensitive only to the splitting

of the excited state. Coherent scattering, on the other hand, is sensitive to both ground

and excited state splitting. The QB in this case occurs between frequencies corresponding

to all allowed transitions between ground and excited states. The quadrupole interaction

on 57Fe splits only the excited state into two sub-levels. Therefore, in both coherent and

incoherent scattering channels a single frequency QB occurs only. However, magnetic

interaction which splits both excited and ground states leads to different QB structures

for the coherent and incoherent scattering [BCR+96].

It was shown in the previous chapter how spatial relaxation influences NFS. Incoherent

scattering, in contrast, is independent of the motion of the nuclei. One can explain this

as follows. The general property of the coherent scattering implies the same energy of the

photon before and after the excitation. The translational diffusion gives rise to a shift of

the nuclear energy during the scattering process. This shift is seen as an additional decay

of the coherent scattering. The incoherent scattering is not sensitive to the energy of the


Table 4.1: Comparison of NFS and incoherent NRS.

NFS Incoherent NRS

Elastic/inelastic scattering Elastic or quasi-elastic

scattering, ∝ fLM2

Inelastic scattering in

most cases, no dependence

on fLM

Direction of scattering Well collimated in forward

direction

Scattering in 4π

Influence of dynamics Spatial dynamics + spin

dynamics (if hyperfine

splitting exists )

Only spin dynamics (if hy-

perfine splitting exists)

Shape of the intensity Combination of natural de-

cay, QB and DB

Natural decay modulated

by QB

photon and therefore is not sensitive to the translational diffusion.

4.2 Principle of SRPAC

Incoherent NRS can be applied to study hyperfine interactions. In this sense it belongs

to the family of nuclear spectroscopy techniques, such as Nuclear Magnetic Resonance

(NMR), Mossbauer Spectroscopy, NFS and Time Differential Perturbed Angular Corre-

lation of γ-rays (TDPAC). Incoherent NRS can be considered as a scattering variant of

the last method and can be called SR-based Perturbed Angular Correlation (SRPAC).

The probability of emission (absorption) of electromagnetic radiation depends, in gen-

eral, on the angle between the expectation value of the angular momentum vector of the

radiating system and the direction in which the radiation is observed. For an ordinary

source of radiation, consisting of an ensemble of many radiating nuclei with their spins

oriented at random, the emission of radiation is isotropic in space. In order to observe

an anisotropic radiation pattern, the distribution of spins in the ensemble must favor a

definite direction or directions, i.e. the ensemble of nuclear spins has to be oriented in

space. Such an ensemble may be prepared by different methods [SA75]:

1. Orientation by observing a preceding emitted radiation in a well defined direction.

This situation is realized in TDPAC.

2. Orientation by absorption of electromagnetic radiation of well defined direction and

well defined polarization. This situation is realized in SRPAC.

At the second step, a nucleus thus oriented emits radiation anisotropically. Therefore

there is an angular correlation between those photons absorbed/emitted at the first step

4.2. Principle of SRPAC 41

γ1

γ2

-, +,EC

TDPAC

γ1 γ2

SRPAC

2nd excited state

1st excited state

excited state

ground state

ground state

γ1 γ2

sample(γ-source)

counterstart stop

detec

tor 1

detector 2

γ1

γ2 detector

SR

start stopcounter

sample

Figure 4.2: The schematic drawing of the principle and experimental setup of TDPAC

(left side) and SRPAC (right side).

and those emitted at the second step. A schematic drawing for SRPAC and TDPAC is

shown in Fig. 4.2.

In TDPAC (left side of Fig. 4.2) a nucleus converts to the so-called second excited state

by β-decay or by an electron capture process from the mother isotope with a standard

lifetime ranging from a few hours to several days. During de-excitation of the second

excited state the photon γ1 is observed by the first detector, and the nucleus goes to the

first excited state. The scattering of γ2 emitted during de-excitation of the first excited

state depends on the angle between γ1 and γ2. Interferences between sub-levels of the

excited state lead to time precession of the angular probability for emission of γ2. The

number of the γ-quanta observed by the second detector as a function of the coincidence

time, e.g. the time between γ1 and γ2, gives the TDPAC intensity as a function of

time and angle between the detectors. More information about TDPAC can be found in

reviews [But96, Mah89].


In SRPAC (right side of Fig. 4.2) a nucleus gets excited via absorption of a SR photon.

Directional selection and timing by the first detector in TDPAC are replaced in SRPAC

by direction, polarization and timing of the incident SR flash.

The angular distribution of radiation depends on the orientation of the nuclear spin

at the time the radiation is emitted. In many experimental situations the time elapsed

between the formation of the oriented state and the time of emission of radiation is long

enough to cause an appreciable change of the orientation of the ensemble of nuclear spins

by hyperfine interactions. It leads to a time structure of the angular correlations, the

shape of which depends on the type of perturbations:

1. Static interactions are caused by the coupling of the nuclear spins with static hy-

perfine fields, i.e., fields which are constant in magnitude and direction during the

lifetime of the excited nuclear state. A constant field causes a precession of the

nuclear spins that result in a periodic behavior of the radiation pattern. It is im-

portant that static interactions will not destroy the orientation of an ensemble, no

matter how long the ensemble is in the excited state and no matter how strong the

fields are.

2. Time-dependent (relaxation) interactions are caused by fluctuating fields, such as

the fields experienced by nuclei in a liquid environment. It may result in a complete

loss of orientation with time.

4.3 Theory of SRPAC in the static case

The theoretical formulation of the angular distribution and correlation of γ-rays has been

developed by Frauenfelder, Steffen and Alder [FS65, SF64, SA75]. We apply their results

to derive the expression for the SRPAC intensity.

A schematic drawing of the experiment is shown in Fig. 4.3. The incoming beam with

wave vector kin is polarized in the horizontal plane which is the plane of the SR storage

ring with polarization vector σ. The scattered beam observed by the detector has the

wave vector kout and the detector covers the spherical angle ∆Ω. The SRPAC intensity,

i.e. the intensity observed by the detector, is given as

I(t) = I0 · e−t/τ0

∫

∆Ω

dΩdW (kinσ,kout, t)

dΩ(4.1)

where τ0 is the lifetime of the excited state, I0 is the intensity of the nuclear fluorescence

in the entire solid angle at zero time and dW (kinσ,kout, t)/dΩ is the differential angular

probability of scattering. The calculation of this function is done in Appendix C. The

differential angular probability of scattering depends, in general, on the multipolarity of

4.3. Theory of SRPAC in the static case 43

Z

YX

Jσ

kin

kout

j

Figure 4.3: A schematic drawing of the SRPAC experiment. The incoming synchrotron

radiation has the wave vector kin and polarization vector σ in the plane of the storage

ring (horizontal plane). The scattered radiation with wave vector kout is observed by the

detector without polarization analysis. The polar and azimutal angles ϑ and φ define the

direction of scattering in the coordinate system connected with the vertical axis.

the photon, the values of the spins for the excited (Ie) and ground (Ig) states of the

nucleus and the direction of the hyperfine field. We limit our consideration here to the

case of 57Fe as a resonant nucleus. Therefore the multipolarity of the photon is M1 and

the spins of the excited and ground states are Ie = 3/2 and Ig = 1/2. The hyperfine fields

are assumed to be isotropically distributed over the nuclear ensemble. It is convenient

to choose a coordinate system where the z-axis is connected with the vertical direction,

i.e. the direction perpendicular to the plane formed by wave vector and polarization of

incoming photon (see Fig. 4.3). Then the angular probability is given as (see eq. (C.13))

dW (ϑ; t)

dϑdφ=

1

4π·(

1 − 1

2P2(cosϑ)G22(t)

)

(4.2)

where ϑ and φ are the polar and azimuthal angles of scattering in the coordinate system

connected with the vertical axis, P2(x) is a Legendre polynomial of the second order, and

G22(t) is called the perturbation factor of second order. G22(t) does not depend on the

geometry of the experiment and contains all the information about the dynamics of the

process. It strongly depends on the type of the hyperfine interaction. If no interaction

exists then G22(t) = 1.

Inserting eq. (4.2) to eq. (4.1) one obtains the expression for the SRPAC intensity

I(t) = I0∆Ω

4π· e−t/τ0 (1 − A22G22(t)) (4.3)


Figure 4.4: The anisotropy coefficient A22 as a function of the polar angle ϑ.

where

A22 =1

2

∫

∆Ω

dΩ

∆Ω· P2(cosϑ) (4.4)

is called the anisotropy coefficient of second order. In this expression the dynamical and

geometrical features of SRPAC are separated into G22(t) and A22. The intensity consists

of two parts: an isotropic part which exponentially decays in time and an anisotropic part

the decay of which is modulated by the perturbation factor. The anisotropy coefficient

A22 is the relative weight of the anisotropic part at zero time.

The anisotropy coefficient A22 depends on the polar angle ϑ. The dependence is

shown in Fig. 4.4 for the case of a point size detector. In the vertical direction, A22 has

a maximum value and decreases with increasing angle. For ϑ ≈ 54.7o the anisotropy

coefficient becomes zero, which means that independent on the shape of the perturbation

coefficient, the SRPAC intensity follows a natural decay in time. In the horizontal plane

(ϑ = 90o) A22 is negative, and its absolute value is equal half the value in the vertical

direction. The time evolution of the anisotropy A22G22(t) observed in the horizontal plane

is an inverse image of that observed in the vertical direction scaled by a factor of two,

independent of the type of interaction.

In the case of static interactions, the perturbation factorG22(t) can be written as [But96]:

G22(t) =1

5+∑

m6=m′

(

3/2 3/2 2

m′ −m N

)2

cos(Em − Em′)t

~(4.5)

where Em denotes the energy of the m-th sublevel of the excited state and the brackets is

the standard notation for the Wigner 3j-symbol. The time evolution of the perturbation

factor in the static case is described by periodic oscillations and an additional constant

term called hardcore.

4.3. Theory of SRPAC in the static case 45

We limit our consideration to the quadrupole hyperfine interactions (see Appendix B).

In this case the expression for the perturbation factor simplifies to

G22(t) =1

5+

4

5cos Ωt (4.6)

where Ω denotes the quadrupole splitting. Using this expression for the perturbation

factor and assuming a point size of the detector in eq. (4.4), the SRPAC intensity can be

written as

I(t) ∝ e−t/τ0 (1 +K cos Ωt) (4.7)

where

K =4A22

A22 − 5=

12 cos 2ϑ+ 4

3 cos 2ϑ− 39(4.8)

Here we introduce the experimentally observed value K - contrast of the quantum beats.

One can see that the time evolution of the SRPAC intensity shows a natural decay

modulated by a single frequency QB. It is similar to the time evolution of the NFS intensity

in the kinematical approximation (see eq. 3.14). However, whereas the contrast of the

QB is equal to 1 for NFS, in SRPAC the contrast is less than 1 and changes with angle.

The simulation of the SRPAC intensity for various polar angles ϑ is shown in Fig. 4.5.

The contrast K is about −0.44 for the vertical direction and about 0.19 for the horizontal

plane. Additionally the NFS intensity is shown in this figure. One can see that the QB

in NFS is in anti-phase to that for SRPAC in the vertical direction and in phase in the

horizontal plane.

One can compare the result for the SRPAC intensity on 57Fe with the TDPAC angular

correlation intensity on the same isotope. The mother isotope is 57Co and two photons

with multipolarity M1 are emitted during the following cascade of spins: 5/2 → 3/2 →1/2. Then the angular probability is given as (see eq. C.14)

dW (ϑ′; t)

dϑ′∝(

1 +1

20P2(cosϑ

′)G22(t)

)

(4.9)

where ϑ′ is the scattering angle defined by kin and kout and G22(t) has the same shape

as for SRPAC. Comparing this expression with the one for SRPAC (see eq. (4.2)), one

can notice that the anisotropy coefficient for TDPAC is 10 times less than for SRPAC.

Also the symmetry axis for the scattered radiation changes from the z-axis for SRPAC

to the y-axis for TDPAC. The maximal value of contrast K for TDPAC is obtained in

the forward direction and equals ∼ 0.04. Such a rather small contrast limits the TDPAC

experiment with 57Fe. Only test measurements have been performed [HRBL69].


Figure 4.5: Simulation of the SRPAC intensity for various polar angles ϑ. In the bottom

the kinematical approximation of the NFS intensity is shown. The quadrupole splitting

was taken as ~Ω = 24Γ0.

4.4 Influence of spin dynamics on SRPAC

Here we consider the influence of time-dependent hyperfine interactions on SRPAC.

This general task will be reduced to the following. We consider a molecule in which

a quadrupole interaction between nucleus and the surrounding exists and the direction

of the EFG, i.e. the spin quantization axis, is defined by the specific molecular struc-

ture. The molecular rotation leads here to a rotation of the EFG. The time evolution of

the perturbation factor is driven by two processes: precession around the EFG with the

characteristic frequency Ω and stochastic rotation of the EFG direction with a charac-

teristic jump rate λ. The ratio of Ω and λ defines the characteristic shape of G22(t). If

the relaxation is slow, i.e. λ ¿ Ω, the time evolution of G22(t) is mainly described by

precession around the EFG which leads to oscillations in time. The slow averaging due

to the stochastic rotation damps the oscillations. In the opposite case, when λ À Ω, the

change of the EFG direction is so fast that no precession appears. The ensemble of spins

relaxes to the isotropic state exponentially.

In general, the calculation of the perturbation factor in the presence of relaxation is

a quite complicated task. There are several ways to solve it using different approaches.

4.4. Influence of spin dynamics on SRPAC 47

One can mention works of Sillescu [SK68, Sil71] as applied to NMR, Afanas’ev and Ka-

gan [AK64] as applied to MS, Blume and Dattagupta [Blu68, Dat81, Dat87] as applied

to MS and TDPAC, Winkler [WG73, Win76] and Lynden-Bell [LB71, LB73] as applied

to TDPAC.

To calculate the expression for the perturbation factor G22(t) in SRPAC we use the

formalism of Blume [Blu68], Lynden-Bell [LB71, LB73] and Winkler [WG73, Win76]. This

formalism is based on the stochastic theory approach. In this approach one separates the

full system into the probe (nucleus) and its surroundings that can be replaced by an

effective medium called heat bath governed by stochastic thermal motions. Heat bath

and probe are connected by hyperfine interaction. Therefore the whole heat bath can be

described by one parameter (in our case the direction of the EFG), the time evolution of

which is governed by the stochastic motion. This stochastic motion is assumed to be a

stationary Markov process, i. e. no memory effect exists. For a precise definition of the

Markov process see e.g. [vK81]. Then, in the density matrix formalism (the definition of

this formalism can be found in [Blu81]), the evolution of the studied system is described

by the time-evolution operator U(t) [WG73]

U(t) = exp

[(

− i

~H× + R

)

t

]

(4.10)

where H× denotes the Liouville operator built on the Hamiltonian of the quadrupole inter-

action (see Appendix B) and R is the operator of the jump probability for the stochastic

state. The operator H× is applied only to the nuclear state and the operator R only to the

stochastic state of the heat bath. However, due to the matrix form of the operators, it is

not possible to factorize the time-evolution operator into stochastic and nuclear parts. If

it were possibles, then the perturbation factor could be expressed as a static perturbation

factor multiplied by the relaxation function. In reality the quantum mechanical nature of

the quadrupole interaction leads to a more complicated time evolution of the perturbation

factor. This fact is known in literature as non-secular effect [Dat87].

To solve eq. (4.10), one has to introduce a suitable assumption regarding the nature of

fluctuation of the orientation of the molecules in the heat bath, i.e., to give a description

of the operator R. We assume that the rotational relaxation is isotropic and happens

by finite angular jumps. Then the main parameter that defines the relaxation process

is the average time τ during which the molecule resides in a given orientation. The

corresponding frequency parameter is the jump rate λ = 1/τ .

At first we consider the finite angular jump model (FJM) which has been developed

by Ivanov [Iva64] and later by Anderson [And72]. The idea of this model is simple. The

molecule resides in a given orientation for a time τ before an angular jump through an

Euler angle α = (0, α, 0) takes place [BDHR01]. Due to the isotropical character of the


relaxation, the 3-dimensional rotation α here can be reduced to one characteristic jump an-

gle α, which is the second parameter of this model. We consider the matrix element of the

jump probability operator R in the angular momentum representation |J, µκ) which is de-

fined by the normalized Wigner rotation matrix elements [Win76]√

(2J + 1)/8π2D(J)µκ (α).

Then due to the isotropical character of the rotational relaxation

(J ′, µ′κ′|R|J, µκ) = λFJMJ · δJJ ′ · δµµ′ · δκκ′ (4.11)

the jump probability depends only on the total angular momentum J . For FJM the jump

probability is

λFJMJ = −λ(1 − PJ(cosα)) (4.12)

The important limit of this model appears in the case of an infinitesimally small jump

angle α. Then the FJM reduces to the well-known rotational diffusion model (RDM)

introduced by Debye to describe the rotational diffusion of a Brownian particle [Deb13].

This model is defined by one parameter, the rotational diffusion coefficient d as [Win76]

λRDMJ = −J(J + 1)d (4.13)

The FJM in the limit of small α reduces to

λFJMJ −−→

α→0−λ · J(J + 1)

(α

2

)2

(4.14)

which makes a connection between the models by

d = λ(α

2

)2

(4.15)

The rotational diffusion model presents an extreme case of the rotational relaxation. The

opposite case is presented by the strong collision model (SCM). In this model the angular

jump has a random magnitude so that after the jump the molecule can find itself oriented

in any direction independent of the previous state. The jump probability matrix element

for SCM can be written as [Win76]

λSCMJ = −λ(1 − δJ0) (4.16)

One can see that except zero angular momentum jumps, the probability is independent

of angular momentum J . This is in contrast with FJM and RDM, where λJ strongly

depends on J . Combining all three models one can write

(J ′, µ′κ′|R|J, µκ) = −λJ · δJJ ′ · δµµ′ · δκκ′ (4.17)

where the specific shape of λJ is defined by the model. The physical meaning of λJ can

be understood if we introduce the rotational correlation function CJ(t) [Dat87, BDHR01]

CJ(t) =∑

µ

< D(J)∗

0µ (α(t))D(J)0µ (α(0)) >=< PJ(cos(α(t) − α(0))) > (4.18)


where the isotropical character of the relaxation is taken into account. This function is

important in many experimental techniques, e.g. nuclear magnetic resonance, dielectric

spectroscopy. Due to the Markov property of the rotational relaxation this function is

exponential and can be written as [BDHR01]

CJ(t) = exp(−λJt) (4.19)

One can see that the quantity λJ which appears in the calculation of the perturbation

factor is the relaxation rate of the rotational correlation function of angular momentum J .

The time-evolution operator U(t) in the shape of eq. (4.10) together with the defined

model for the rotational relaxation gives a formal answer to the question about the time

evolution of the perturbation factor. But there is a still open question about the practical

solution of this equation. There are several ways to do it: resolvent method [Dat87],

series expansion method [WG73], eigensystem method [LB73, Win76]. Also one should

notice that not for every model there is an analytical solution for the perturbation factor.

For the straightforward numerical calculations we found quite convenient the eigensystem

method which has been developed by Winkler in application to the TDPAC with spin

I = 5/2 [Win76]. The procedure of the calculation of the perturbation factor G22(t)

for the spin I = 3/2 is presented in Appendix D. One can notice that in our case the

eigensystem problem reduces to the diagonalization of a 6 × 6 matrix. Also it is seen

that the influence of the rotational relaxation to the perturbation factor reduces to 2

parameters λ2 and λ4 which are the relaxation rates of the second and the forth angular

momentum correlation functions. Since both λ2 and λ4 are proportional to the jump rate

in the considered models, it is convenient to replace them by another pair of parameters

λ2 and q ≡ λ4/λ2. The parameter q does not depend on the jump rate λ and describes

the model of the relaxation. It is equal to 1 for SCM and 20/6 for RDM. The dependence

of q on the jump angle α in FJM is shown in Fig. 4.6.

In general there is no analytical presentation of G22(t). However, for q = 1 (SCM)

an analytical answer can be obtained. For this purpose the resolvent method which was

developed by Dattagupta is quite convenient. In Appendix E we obtain the analytical

expression for G22(t) in the SCM using this method.

As it was mentioned at the beginning of this section, the time evolution of the per-

turbation factor strongly depends on the ratio between the characteristic jump rate and

the quadrupole splitting. Therefore we can separately consider the behavior of G22(t) in

three regimes: the slow relaxation regime where λ2 ¿ Ω, the fast relaxation regime where

λ2 À Ω and the intermediate regime where λ2 ∼ Ω.


Figure 4.6: The parameter q as a function of jump angle α in the FJM. Additionally, the

values of q for the SCM and the RDM are shown.

4.4.1 Slow relaxation regime

At first we consider the slow relaxation regime. The simulation of G22(t) for different

values of λ2 and q is shown in Fig. 4.7. In the case of q = 1 (SCM) the analytical

approximation of the perturbation factor for λ2/Ω ¿ 1 can be written as (see eq.(E.19))

GSCM22 (t) ' 1

5e−4λ2t/5 +

4

5e−3λ2t/5 cos(Ωet− φ) (4.20)

Ωe = Ω

(

1 − 4λ22

25Ω2

)

(4.21)

φ =4λ2

5Ω(4.22)

One can see that the expression for G22(t) is similar to the static case (see eq.(4.6)) but due

to the relaxation both terms are exponentially damped. Additionally the QB changes its

frequency and phase. The damping rate is proportional to λ2 with different proportionality

coefficients for the hard core and for the oscillatory term. This is a consequence of the

non-secular effect discussed above.

The time evolution of the spin is mainly governed by the precession around a quan-

tization axis, so that the scattered signal is a periodic oscillation. The quantization axis

is defined by the direction of the incoming radiation at time zero. With increasing time

the orientation of the axis becomes distributed due to the molecular rotation. As result

the amplitude of the oscillation decreases with time. The decrease is described by the

correlation between the orientation at time zero and at time of reemission. In our case this


Figure 4.7: Simulation of the perturbation factor G22(t) in the static case and for various

small values of λ2. The solid line corresponds to q = 0.2, the dashed line to q = 1 (SCM)

and the dotted line to q = 20/6 (RDM).

correlation function is an exponential function with rate proportional to λ2. Therefore,

the perturbation factor is the static perturbation factor describing the spin precession

multiplied by an exponential decay describing the effect of rotation. However, due to the

quantum mechanical nature of the spin variable, the change of the direction of the quan-

tization axis additionally results in a mixing of the population of the spin states. This

leads to different values of the exponential rates for the two terms in the perturbation

factor. While both of them are proportional to λ2, their ratio gives information about the

characteristic jumps (model of the reorientation).

It is shown in Appendix D that in the general case the slow relaxation approximation

of G22(t) can be written as

G22(t) '1

5e−pht +

4

5e−pot cos(Ωet− φ) (4.23)

where the damping coefficients depend on q and λ2 as

ph = λ2

(

2

7+

18

35q

)

(4.24)

po = λ2

(

75 + 37q −√

1009q2 − 450q + 225)

/140 (4.25)


Figure 4.8: The values of ph/λ2 and po/λ2 obtained from the fit of the perturbation

factor by eq. (4.23). The perturbation factor was simulated by the procedure described in

Appendix D with different values of λ2 and q. The left side shows the linear dependence

of ph and po on λ2 up to λ2 . 0.3. The right side shows the dependence of ph/λ2 and

po/λ2 on q. The lines show the approximation of eqs. (4.24),(4.25).

The range of parameters where this approximation is correct was checked by fitting the

numerical simulation of G22(t) according to Appendix D by eq. (4.23) with free parameters

ph and po . The obtained ph and po, scaled by λ2, are shown in Fig 4.8 as functions of

λ2 and q. The solid lines show the dependences obtained by eqs. (4.24),(4.25) . One can

see that in the region of λ2/Ω < 0.3 and 0.2 < q < 5 the approximation (4.23-4.25) works

quite well. As a consequence one can say that the measurement of the damping of hard

core and oscillatory term are necessary to obtain information about the relaxation rate

and the mechanism of the relaxation. The ratio ph/po is independent on λ2 and is defined

by the model of the rotational relaxation.

4.4.2 Fast relaxation regime

The fast relaxation regime is defined by λ2/Ω À 1. The simulation of the perturbation

factor is shown in Fig. 4.9. One can see that the oscillations disappear and the time

evolution of G22(t) is described by an exponential decay with a decay rate that decreases

with increasing λ2.


Figure 4.9: Simulation of the perturbation factor for various high values of λ2/Ω.

The general solution for SCM reduces in the fast relaxation regime to (see eq. (E.20))

GSCM22 (t) ' exp

(

−4Ω2

5λ2

t

)

(4.26)

The nucleus finds itself at time zero in a state with a certain direction of the quantization

axis. Its spin starts to precess around this axis, but, since the jump rate is very fast, it

can hardly precess at all before the axis jumps to a new direction. Since the nuclear spin

can not follow such a rapid relaxation it sees an averaged EFG. With increase of the jump

rate the averaged EFG decreases and therefore the rate of the spin motion decreases as

well. As result the loss of the anisotropy which is produced by the spin motion slows

down. In the case of an infinitively large value of the jump rate the ensemble of spins is

frozen in the initial position and the perturbation factor stays at constant value.

The eq. (4.26) is consistent with the general result obtained by Abragam and Pound [AP53]

in the fast relaxation regime, also called Abragam-Pound limit. For perturbations gov-

erned by quadrupole interaction and assuming that the relaxation does not change the

absolute value of the EFG, one can write the perturbation factor for spin I = 3/2 in the

fast relaxation regime as (see eqs. (13.291) and(13.294) in [SA75])

G22(t) ' exp

(

−4

5Ω2 < τ2 > t

)

(4.27)

< τ2 > =

∫ ∞

0

dt · C2(t) (4.28)


where < τ2 > is the usual definition of the mean relaxation time for the rotational cor-

relation function C2(t). Taking into account the expression for C2(t) (see eq. 4.19) one

obtains that < τ2 >= 1/λ2 which reduces eq. (4.27) to eq. (4.26). The general expression

for G22(t) is valid even in models of non-exponential decay of the rotational relaxation,

like the stretched exponential relaxation with correct definition of < τ2 >. However, the

dependence of G22(t) on λ4 which was observed in the slow relaxation regime disappears

in this regime. Therefore the information about the type of relaxation can not be obtained

in the fast relaxation regime.

4.4.3 Intermediate relaxation regime

The intermediate regime is characterized by the relation λ2 ∼ Ω. The behavior of the

perturbation factor changes in this regime from damped oscillations to an exponential

decay. The simulation of the perturbation factor in this regime is shown in Fig. 4.10 for

various values of λ2 and q.

It is shown above that the appearance of the rotational correlation function in G22(t) in

the slow and fast relaxation regime is completely different. Whereas for the slow relaxation

regime the correlation function is observed straightforward in time (taking into account

the non-secular effect), only the integral of this function appears in the perturbation factor

in the fast relaxation regime. In this sense we can consider the intermediate regime as a

regime where the slow relaxation behavior is already overdamped and the fast relaxation

behavior does not yet appear. Such a point of view corresponds to the fact that only

the rather short-time region is important at this regime. The long-time tail of G22(t)

approaches zero.

4.4.4 Short-time dynamics in restricted anglular range

Till now we considered a long-time (structural) relaxation that describes an irreversible

dynamical processes. Now we consider the influence of fast rotational dynamics on the

time evolution of the perturbation factor. This dynamics is presented by a rotation of

the molecule in a restricted angular range around the equilibrium position, the libration.

This process is reversible and qualitatively differs from the long-time dynamics. The effect

of the short-time dynamics on G22(t) can be calculated precisely in the fast relaxation

regime. An additional prefactor reflecting the decrease due to the fast librations appears

in the rotational correlation function C2(t). This prefactor, which we call frot, is equivalent

to the Lamb-Mossbauer factor and can be calculated as

frot =< P2 (cos(α(t) − α(0))) >=< P2 (cos(α(t))) >=1

4+

3

4exp

[

−2 < α2 >]

(4.29)


Figure 4.10: Simulation of the perturbation factor in the intermediate relaxation regime.

The solid line corresponds to q = 0.2, the dashed line to q = 1 (SCM) and the dotted line

to q = 20/6 (RDM).

where we assume that α(0) = 0 is the equilibrium position and where we use for the

angular motions the relation [LFC02]

< eıcα >= exp

[

−1

2c2 < α2 >

]

(4.30)

< α2 > is the mean square angular displacement from equilibrium. Inserting this prefactor

into C2(t) in eq. (4.28), we obtain G22(t) in the fast relaxation regime as

G22(t) ' exp

(

−4Ω2frot

5λ2

t

)

(4.31)

where frot describes the short-time dynamics and λ2 describes the long-time final decay

of the correlation function. Physically, the reversible uncorrelated in time fast angular

motions in a restricted angle lead to an average of the quadrupole interactions over this

angle. Therefore the influence of the short-time dynamics can be reduced to the scaling

of the quadrupole splitting value as Ω = Ω0

√frot where Ω0 is the static value of the

quadrupole splitting. We assume that this result is valid not only in the fast relaxation

regime but in the slow relaxation regime as well. Taking into account small values of α


one can write

Ω = Ω0 ·(

1 − 3

4< α2 >

)

(4.32)

A similar result was obtained in NMR [BDHR01] and in electron spin resonance [Dzu96].

4.5 Influence of spatial and spin dynamics on NFS

4.5.1 Influence of spin dynamics

In Chapter 3 we have considered the influence of spatial dynamics on NFS. It was pointed

out there that also time-dependent hyperfine interactions lead to a modification of the time

evolution of NFS. Here, we present the theoretical description of this influence. Like in the

case of SRPAC we consider a molecule in which a quadrupole interaction between nucleus

and surroundings exists and where the direction of the EFG is defined by the molecular

structure. The time evolution of the quadrupole interaction is defined by the rotation of

the molecule and, respectively, by the rotation of the EFG. The approach used to explain

the influence of the rotation on SRPAC can also be used in application to NFS. One can

introduce the perturbation factor G(t) which describes the time evolution of the density

matrix, as done in eq.(2.27) of [Dat81]. The difference with respect to SRPAC is that in

SRPAC only the perturbation of the excited state is considered whereas NFS is influenced

by the simultaneous perturbation of excited and ground states. Correspondingly, the

NFS density matrix is formed by both excited and ground states. Also by contrast to

SRPAC the perturbation factor describes the time evolution of the amplitude of scattering.

Nevertheless the time evolution of the density matrix is defined by the same eq. (4.10),

only the action of the Liouville operator representing the quadrupole interaction on the

density matrix will be different.

The self-correlation function K(t) of NFS which was introduced in eqs. (3.4) and (3.16)

is given as

K(t) = e−t/2τ0 ·G(t) (4.33)

This function also gives the time evolution of the NFS amplitude in the kinematical

approximation.

In the static case the quadrupole interaction results in a simple expression for the

perturbation factor (see eq. (3.13))

G(t) = cos(Ωt/2) (4.34)

The influence of rotation on G(t) has been calculated in [DB74, Dat76] for the RDM and

the SCM by the resolvent method. The result was obtained as a Laplace image G(p).

4.5. Influence of spatial and spin dynamics on NFS 57

Making the inverse Laplace transformation we obtain the expression for G(t) which can

be written for both models as

G(t) = e−λ2t/2 cos

(

Ωet

2− φ

)

(4.35)

φ = arctan

(

λ2

Ωe

)

(4.36)

Ωe = Ω(

1 − λ22/Ω

2)1/2

(4.37)

where λ2 is the same as in SRPAC and is defined by the model of rotation as in the previous

section. The corresponding expression for the NFS intensity in kinematical approximation

is given as

I(t) = I(0) · e−t/τ0 · e−λ2t · (1 + cos(Ωet− 2φ)) /2 (4.38)

The time evolution of the perturbation factor and the NFS intensity are shown in Fig. 4.11

for several values of the relaxation parameter λ2. The time evolution of the NFS intensity

in the static case is defined by a natural decay modified by a QB. This time evolution is

similar to that of SRPAC with the contrast K = 1 in eq. (4.7).

Similar to SRPAC, the time evolution of the NFS intensity behaves differently for

small and large values of the relaxation rate λ2 - slow and fast relaxation regimes can

be defined. In the slow relaxation regime, the time evolution is characterized by a QB

damped in time. When λ2 becomes larger than Ω, Ωe becomes complex and the cosine

function goes to a hyperbolic cosine. The QB is not seen any longer, and in the fast

relaxation regime, where λ2 À Ω, eq. (4.38) is approximated by an exponential decay

I(t) ' I(0) · e−t/τ0 · exp

(

− Ω2

2λ2

t

)

(4.39)

One can see that the influence of spin relaxation on SRPAC and NFS is similar but

differs in details. The SRPAC intensity consists of isotropic and anisotropic contributions,

the first one does not depend on relaxation. On the other hand, NFS includes only

an anisotropic contribution and all intensity is damped due to relaxation. The NFS

perturbation factor does not include a hard core term, only a QB term is present. As

result, one can not get information about the model of rotation, which was possible in

SRPAC from comparison of the damping rate of the hard core and QB terms. Besides

that, the effective frequency Ωe of the QB which decreases relative to Ω due to relaxation,

decreases differently in SRPAC and in NFS. As a result, different QB frequencies are seen

in NFS and SRPAC.

Eq. (4.39) gives the kinematical approximation for the NFS intensity. The general

expression can not be obtained analytically. The procedure developed in Appendix A has

to be used in order to take into account multiple scattering. However, the exponential


Figure 4.11: Simulation of the time behavior of the perturbation factor and of the NFS

intensity in kinematical approximation for several values of the relaxation rate λ2. The

quadrupole splitting was chosen as ~Ω = 24Γ0.

character of the damping of G(t) in both the slow and the fast relaxation regime leads

to factorization of the multiple scattering and damping terms similar to eq. (3.19). The

cosine part of G(t) in the slow relaxation regime can be extracted from the multiple

scattering correction in the way presented in eq. (3.15). However, this approximation is

valid for ξ/Ωeτ0 < 1. In the intermediate relaxation regime, where λ2 ∼ Ω, Ωe ∼ 0 and

the last approximation is not valid. Then the intensity has to be calculated numerically.

4.5.2 Influence of spin and spatial dynamics

It was assumed till now that the molecules experienced only one type of motion: transla-

tion or rotation. In general one has to consider the time evolution of the nuclear ensemble

governed by both types of motion. This task has been solved by Dattagupta [Dat75,

Dat76] who applied the following model of relaxation. The nucleus resides in a given

state for a time τ . The corresponding relaxation rate is λ = 1/τ . It jumps instanta-

neously to a new site in space and in angle. The angular jump is described by the model

4.5. Influence of spatial and spin dynamics on NFS 59

of rotation which is discussed in Section 4.4 and is defined by λ2. The jump in space is

introduced in a similar way. The spatial analogue of the FJM represents the jump of the

molecule over a distance a. The corresponding jump probability in the momentum space

can be introduced as [Dat75]

λt(k) = λ

(

1 − sin ka

ka

)

(4.40)

This value represents the decay rate of the exponential van-Hove self-intermediate function

Fs(t). The momentum k is fixed by resonance energy, k = 7.31 A−1 for 57Fe, and in the

following we omit the dependence of λt on k.

The eq. (4.40) simplifies in the hydrodynamical approximation where a is assumed to

be small. Then

λt ' λk2a2/6 = Dk2 (4.41)

where the diffusion coefficient D is introduced. This approximation is similar to the RDM

for the rotational motion.

The model presented above depends on three parameters which describe the jump

frequency and the characteristic jump angle and distance. We introduce these parameters

via λ, λ2 and λt. The NFS intensity in the kinematical approximation can be obtained

from [Dat75, Dat76] by Laplace transformation and proper replacement of the variables

that results in

I(t) = I(0) · e−t/τ0 · e−2λNFSt (1 + cos(Ωet− 2φ)) /2 (4.42)

λNFS = λt + λ2/2 (4.43)

φ = arctan

(

λ2

Ωe

)

(4.44)

Ωe = Ω

(

1 − λ22

Ω2

)1/2

(4.45)

λ2 = λ2

(

1 − λt

λ

)

(4.46)

This set of equations is similar to the one obtained for pure rotational relaxation (see

eqs. (4.35)-(4.38)). The difference is an additional exponential damping due to the trans-

lational relaxation and the effective decrease of λ2 to λ2. This decrease can be explained

as follows. The spatial motion of the molecule leads to the cancelation of the contribu-

tion of the corresponding wavelet to the coherent scattering. Therefore, any subsequent

molecular rotation has to be excluded from the consideration.

The main feature of the NFS intensity which will be considered in the experiment is

the additional decay with decay rate 2λNFS. This parameter can be rewritten as

2λNFS = λ2 + λt

(

2 − λ2

λ

)

(4.47)


The value of λ2/λ depends on the model of rotation. For the two extremal cases of the

RDM and SCM it is equal to 0 and 1. Therefore, the coefficient of λt can vary in the range

from 2 to 1 depending on the model of rotation. We can assume that the variation of

λt with temperature occurs much faster than the temperature variation of the rotational

model. As result, we can say that the additional decay rate of the NFS intensity is formed

by sum of the rotational relaxation rate λ2 and the effective translational relaxation rate

λt which is proportional to the real rate of translation λt with the scaling coefficient

between 1 and 2. In the following we will make no difference between λt and λt and will

use following equation

2λNFS = λ2 + 2λt (4.48)

with corresponding definition of λt. Also it is important to notice that for any model of

rotation 2λNFS in eq. (4.47) will be larger than the sum of λ2 and λt. If λ2 derived from

SRPAC is equal to 2λNFS derived from NFS it unambiguously means that λt = 0.

The presented result was derived in the model where both translational and rotational

dynamics are defined by the same jump frequency. However, the eq. (4.48) is valid also in

the general case of the different frequencies. The jump frequency λ is significant for the

rotational dynamics since the time evolution is described by the new operator after each

jump. On the other hand, the operation of the spatial jump commutates with the time

evolution. Therefore, all equations above are valid also for the case when the translational

dynamics is defined by other frequency λ′. Here, the eq. (4.41), for example, is rewritten

as

λt = λ′k2a2/6 = λ · λ′

λk2a2/6. (4.49)

where (λ′/λ)k2a2/6 is the effective jump probability increased/decreased due to the dif-

ference in the frequencies.

The generalization of the kinematical approximation of eq. (4.42) which takes into

account multiple scattering can be done in the way discussed in the previous section.

Chapter 5

Experimental

In this chapter the setups for the NFS and SRPAC measurements are introduced and

components are described. Several methodological aspects of SRPAC are discussed: the

influence of the coherent scattering on SRPAC; different ways to obtain pure incoherent

scattering; the dependence of SRPAC on the geometry of the experiment.

5.1 Experimental setup

In both NFS and SRPAC experiments one studies the time dependence of the delayed

scattered intensity following the excitation of a nuclear ensemble by the short X-ray pulse

(typically 100 ps). Both scattering channels can be measured in parallel. In Fig. 5.1 the

typical setup is shown.

The undulator produces an X-ray beam with an energy width ∆E ≈ 300 eV. Radiation

with energy outside a µeV window centered at resonance energy E0 does not undergo

resonant interaction with the nuclei of the target. Therefore this part of radiation should

be eliminated as much as possible in order to reduce the load on the detectors. This

monochromatization is achieved in two steps. A high-heat-load premonochromator (PM)

has the task of handling the heat load of the “white” radiation produced by the undulator

and reducing the energy bandwidth of the radiation to the eV region. This is achieved

with two Si(1 1 1) reflections arranged in a non-dispersive setting [RC96]. A second step

of monochromatization is achieved with a high resolution monochromator (HRM) which

typically reduces the energy bandwidth down to 1 − 10 meV. A nested design of the

HRM was used with four successive reflections on two nested channel-cut crystals. The

Si(4 2 2) reflection for the outer crystal and Si(12 2 2) for the inner one give an energy

bandwidth ∆E ≈ 6.4 meV [IYI+92]. Another setup uses only the outer crystal of the

HRM in reflection Si(8 2 2) which gives ∆E ≈ 150 meV.

After the interaction with the sample, the scattered photons have to be detected. As

61

62 Chapter 5. Experimental

PM

HRM

Al foilUndulator

Sample

APD

APD

Figure 5.1: Typical setup for NFS/SRPAC measurements. The flash of SR light generated

by the electrons in the undulator is monochromatized in two steps by a high-heat-load pre-

monochromator (PM) and a high resolution monochromator (HRM). The signal scattered

by the sample is recorded by fast detectors formed by avalanche photo-diodes (APD). The

NFS signal is observed by a detector in the forward direction, and the SRPAC signal is

observed by a detector below the sample. According to Fig. 4.3 this corresponds to ϑ = 0.

The second detector is covered by an Al foil.

the nuclear scattering has to be separated in time from the electronic one, fast detectors

are required (with time resolution of ∼1 ns). Moreover these detectors should have a fast

recovery time, to be able to stand a rather intense pulse of X-rays (the prompt pulse,

typically of the order of 107 ÷ 109 photons/s) and a few ns later to count the delayed

photons with much lower intensity (typically 10−1 ÷ 103 photons/s). Avalanche photo-

diodes (APD) are normally used for this purpose [BRM97], as they have the required

characteristics together with a very low background (∼ 10−2 Hz).

The NFS signal is recorded by the detector in the forward direction far away from

the sample (∼1 m). As the coherent scattering is well collimated in forward direction no

loss of the signal due to divergence of the beam occurs. At the same time the part of

the 4π halo formed by incoherent scattering does practically not contribute to the signal

recorded by this detector.

The SRPAC intensity is recorded by a detector situated close to the sample (5÷30 mm)

usually in 90o geometry below the sample. The choice of the detector position is defined by

the optimum flux of the useful signal and will be discussed in the next section. The SRPAC

detector is covered by an Al foil. The nuclear resonant fluorescence with 14.4 keV energy of

the scattered photons is accompanied by atomic fluorescence whose probability is 8 times

higher for 57Fe. This process mainly occurs through emission of 6.4 keV photons. This

part of the scattered radiation is eliminated by introducing the Al foil. The attenuation

lengths for these two energies are l6.4 ≈ 40 µm and l14.4 ≈ 440 µm, respectively. An Al foil

with thickness 320 µm transmits ∼ 50 % of the nuclear fluorescence and only ∼ 0.03 % of

5.2. Methodical aspects of SRPAC 63

the atomic fluorescence. This gives us the possibility to neglect the atomic fluorescence

signal.

Standard fast timing electronics [Bar01] is used to process the signal from the APDs.

A reference signal from the radio frequency system of the storage ring provides the syn-

chronization of the electronics with the incoming pulse.

5.2 Methodical aspects of SRPAC

Whereas NFS and NIS were studied extensively during the last decade, SRPAC is a new

method that requires methodological experimental studies. In this section two aspects of

this method are analyzed experimentally: the dependence of the scattering on position

and size of detector and sample and the influence of NFS to SRPAC. The experiments

were performed with a glass former dibutylphthalate doped by ferrocene (DBP-FC), which

will be described in Chapter 5. The nuclei in the sample experience a static quadrupole

interaction which is isotropic over the sample.

5.2.1 Dependence on the geometry of the experiment

It was shown in the theoretical section that the essential part of SRPAC, the anisotropy

A22G22(t), can be extracted from the scattered intensity I(t) as (see eq. (4.3))

A22G22(t) = 1 − I(t)/I0 · et/τ0 (5.1)

where I0 = I0∆Ω/4π can be easily obtained from the fit. The anisotropy is factorized into

the perturbation factor and the anisotropy coefficient. The perturbation factor, G22(t),

describes the time evolution of scattering and does not depend on the geometry of the

experiment. Therefore, the time evolution of the anisotropy is the same in every direction

taking into account the anisotropy coefficient A22. In the following we check this feature

in experiment.

The SRPAC intensity was measured at two temperatures, 200 and 296 K, which cor-

respond for the investigated sample to the slow and fast relaxation regimes, respectively.

For each temperature two measurements were performed with the detector in 90 degrees

to the beam in vertical and horizontal directions. With respect to Fig. 4.3 this corresponds

to the polar angle ϑ = 0 and 90 respectively. The results of the measurements are shown

in Fig. 5.2. At T = 200 K quantum beats are seen in the time evolution of the intensity

which are in antiphase for the horizontal and vertical directions. The perturbation factor

extracted from the intensity is the same for both directions within experimental error.

The corresponding anisotropy coefficients are Aver22 = 0.38 and Ahor

22 = −0.2. They are not

the same as in the ideal case Aver22 = 0.5 and Ahor

22 = −0.25 due to the finite sizes of the


Figure 5.2: Comparison of the SRPAC intensity I(t) (left side) and perturbation factor

G22(t) (right side) on DBP-FC for the detector being at 90o to the incoming SR and

in the horizontal (open circles) and vertical (solid circles) position with respect to the

plane of the synchrotron. The upper part corresponds to T = 200 K where almost static

perturbations are observed, the lower part corresponds to T = 296 K where fast relaxation

is observed.

detector and the sample. At T = 296 K the quantum beats are overdamped. A slow tran-

sition of the signal into a final natural decay at large times is here the characteristic feature

of the intensity recordered by both detectors. This transition occurs via deceleration of

the decay for scattering in the horizontal direction and via acceleration of the decay for

scattering in the vertical direction. The perturbation factor again follows the same time

evolution for both directions. The obtained anisotropy coefficients are Aver22 = 0.38 and

Ahor22 = −0.24. Here the experiments demonstrate that different geometrical conditions

yield nevertheless the same perturbation factor G22(t).

This independence of the perturbation factor of the scattering direction allows one to

put the detector in any position around the sample. Most convenient setup is the vertical

direction since the maximum anisotropy is observed at this scattering angle. This position

of the detector was used in the following experiments.

The second geometrical parameter that appears in the expression for the SRPAC

intensity (eq. (4.3)) is the solid angle ∆Ω subtended by the detector on the sample. If

the size of the detector is fixed then ∆Ω is defined by the distance between the detector


Figure 5.3: The dependence of the anisotropy coefficient A22 (dashed line) and of the

efficiency function R (solid line) on the polar angle ϑc covered by the detector.

and the sample. When the detector is far away from the sample, ∆Ω ∼ 0, the observed

countrate will be close to zero and no SRPAC signal will be detected. In the opposite

case, when the detector is close to the sample, ∆Ω ∼ 2π, the countrate will be maximal

but the anisotropy coefficient is A22 ∼ 0 due to spatial average. Therefore, between these

two extreme cases an optimal angle (distance) exists. The part of the scattered intensity

connected with the perturbation factor in eq. (4.3) is proportional to

R =∆Ω

4πA22 (5.2)

which can be considered as an efficiency function.

For simplification let us consider a detector of cylindrical form with its axis lying along

the vertical axis. Then, the setup will be symmetrical around the vertical axis and the

solid angle ∆Ω can be written as ∆Ω = 2π(1 − cosϑc), where the polar angle ϑc is the

angle between vertical axis and the direction to the border of the detector. Inserting this

condition to the expression for A22 (4.4) we obtain

A22 =cosϑc(1 + cosϑc)

4(5.3)

R =cosϑc(1 − cos2 ϑc)

8(5.4)

The dependence of A22 and R on the angle ϑc is shown in Fig. 5.3. One can see that the

anisotropy coefficient has a maximal value 0.5 for ϑc = 0 and decreases when ϑc increases.

The efficiency R is zero for the extremal angles and shows a maximum at ϑcr ' 54.7o.

This angle corresponds to an anisotropy coefficient A22 ' 0.23.

The obtained optimum angle was calculated assuming a detector of cylindrical form

and a point-like sample. The real experimental situation requires to take into account the


Figure 5.4: The dependence of the anisotropy coefficient A22 and of the efficiency function

R on the distance d between detector and sample for the 5 × 5 mm2 and 10 × 10 mm2

detectors.

sizes of sample and detector. The sample has the surface 10 × 2 mm2 where 2 mm is the

horizontal width of the beam. Detectors of square areas 5×5 mm2 and 10×10 mm2 were

used in the vertical direction. The dependence of A22 and R on the distance d between

detector and sample is shown in Fig. 5.4. One can see that the maximum efficiency

corresponds to the distance 2−5 mm depending on the type of detector. The characteristic

anisotropy coefficient for the optimal distance is around 0.25 which agrees with the simple

case presented in figure 5.3

For the experimental study of the dynamics the detector 10× 10 mm2 was chosen and

the distance between detector and sample was d = 10± 1 mm. The anisotropy coefficient

for these conditions is A22 = 0.38 ± 0.015.

5.2.2 Contributions to 4π scattering produced by NFS

The incoherent nuclear scattering is not the only channel of scattering which produces de-

layed radiation into 4π. Also the combination of NFS and subsequent Rayleigh scattering

can direct delayed radiation into the full solid angle [SK95]. This additional contribu-

tion can be so strong that it completely overshadows the SRPAC signal. Its strength is

proportional to the NFS signal.


Figure 5.5: The probability of the resonance absorption S(δE) as a function of the energy

difference δE between energy of the incoming photon Eγ and resonant energy E0. The

hatched rectangles show the positions of the HRM used in the present experiment.

There are several cases when pure SRPAC is observed. The first and evident case is

when no NFS exists. Since the NFS intensity is proportional to f 2LM , this case is realized

when fLM ∼ 0. By this way one can favorably apply SRPAC to studies of soft condensed

matter. It is particularly suited for investigation of glass formers in the temperature range

where MS and NFS are not applicable.

When fLM is not zero and NFS is dominant, another way to see pure SRPAC has

to be explored. This way had been used first by Baron [BCR+96]. The energy of the

incoming radiation is tuned out of resonance by 10 ÷ 100 meV. The resonant absorption

occurs with creation/annihilation of phonon, which prohibits nuclear coherent scattering.

The glass former DBP-FC is a convenient sample to study how strong can be the con-

tribution into 4π connected with NFS. The Lamb-Mossbauer factor drastically decreases

with temperature so that measuring the incoherent signal at several temperatures one

can obtain completely different contributions of NFS-Rayleigh scattering. Two temper-

atures were chosen: 100 and 200 K. An almost static quadrupole interaction describes

the perturbation of the nuclear ensemble for both temperatures. At the same time, the


Figure 5.6: Comparison of the delayed intensity divided by natural decay measured by

three methods: NFS, SRPAC in resonance and SRPAC out of resonance. The left side

shows data at 100 K (fLM ≈ 0.36) and the right side shows data at 200 K (fLM ≈ 0.004).

The solid lines show fits of the SRPAC intensity according to theory.

Lamb-Mossbauer factor is 0.36 for 100 K and 0.004 for 200 K, i.e. decreases by a factor

of ∼ 100. The energy of the incoming radiation was chosen ”in resonance” and ”out of

resonance”.

The probability of the resonant absorption S(δE) as a function of the energy difference

δE between energy of the incoming photon Eγ and the resonant energy E0 is shown in

Fig. 5.5 for DBP-FC at 100 and 200 K. The peak at δE = 0 corresponds to the elastic

and quasielastic absorption. Its amplitude slows down with temperature. The other part

of S(δE) represents the absorption with creation/annihilation of phonons. A pronounced

feature of the absorption probability is the existence of two peaks at δE ∼ 20 meV and

δE ∼ 60 meV. The hatched rectangles in Fig. 5.5 show the positions of the HRM used in

the experiment. The position ”in resonance” is centered at E0 and covers ∼ 50 % of the

absorption probability. The other positions ”out of resonance” are adjusted to the energy

of the side peaks and cover 1 − 2 % of the absorption probability.

Fig. 5.6 shows for two temperatures three types of scattering which were measured:


NFS, incoherent scattering ”in resonance” and incoherent scattering ”out of resonance”

which is pure SRPAC. The incoherent scattering in resonance is the sum of SRPAC

the strength of which is a constant with temperature and NFS-Rayleigh scattering the

strength of which decreases with temperature as f 2LM . At 200 K, where fLM is small, the

incoherent scattering ”in resonance” consists of pure SRPAC. On the contrary, at 100 K,

the contribution of NFS-Rayleigh scattering is relatively large which is clearly seen in the

figure. However, this contribution mainly appears at early times, below 150 ns in the

figure. This is due to the redistribution of NFS scattering to smaller times for a large

effective thickness. This fact allows us to measure SRPAC ”in resonance” using a time

window which is cut at small times. As it is seen in the figure, the beginning of this time

window has to be where the base line of the QB becomes a constant.

Chapter 6

Study of LGT dynamics by NFS and

SRPAC

This chapter is dedicated to the application of SRPAC and NFS to the study of dynamics

of the LGT. In the first section a review of MS studies of the LGT is presented and

questions, which have to be clarified, are pointed out. The used glass former is described

in Section 6.2. After that the study by SRPAC and NFS is presented and the treatment

of the data is explained. The obtained results are discussed in the next section and

are compared with results of dielectric spectroscopy. The last section is dedicated to

the extraction of information about the stretching of the relaxation function from NFS

measurements.

6.1 Previous studies of the LGT by Mossbauer spec-

troscopy

The sensitivity of MS to dynamics on the time scale up to µs (down to neV on the energy

scale) and on the Angstrom length scale makes it a convenient tool to study relaxation

processes in viscous liquids. This was recognized immediately after the discovery of the

Mossbauer effect [CS63, BEHW63]. However, the nuclear resonance measurements require

a resonant isotope in the sample, for practical purposes this means a decent amount of

iron in the sample. This strongly restricts the selection of suitable systems. The problem

can be overcome by dissolving a small amount of iron-containing molecules in a glass-

forming liquid. Such molecules can be considered as a probe inserted to the glass former,

or as the second component of a binary mixture. This method was widely used to study

dynamics in organic glass formers. One of the often used systems is the ferrous ion

Fe2+ dissolved in a glycerol-water mixture [CS63, AM72, CHR75, NFP91]. Another class

of glass formers was obtained by dissolving ferrocene in organic glass formers: dibutyl

71

72 Chapter 6. Study of LGT dynamics by NFS and SRPAC

10-1

f LM

10-2

10-1

150 200 250

30

33

Temperature T [ K ]

Tg

Ω

[ Γ

0 ]

100 150 20023

24

Temperature T [ K ]

Tg

4 510

-1

100

101

102

∆Γ [

Γ0

]

1000 / T 5.0 5.5

10-1

100

101

Tg

Tg

1000 / T

Figure 6.1: Study of the LGT by MS: Fe2+ in glycerol (left side) studied in [AM72](•) and

in [NFP91](); ferrocene in dibutyl phthalate [RZF76] (right side). The Lamb-Mossbauer

factor fLM (top) and the quadrupole splitting ~Ω (middle) are shown as functions of

temperature, and the line broadening ∆Γ (bottom) is shown as a function of inverse

temperature. The data for ∆Γ and ~Ω were scaled by Γ0 = 0.097 mm/s. fLM for

Fe2+ in glycerol is obtained by scaling data from [AM72] to the known value of fLM at

130 K [Chu03].

phthalate [RZF76] and o-terphenyl [VF80]. Also, MS was applied to study dynamics of

iron-containing polymers [GHL+98] and biomolecules [CHK+96].

Three pronounced features of MS data can identify the LGT: an anharmonic de-

crease of the Lamb-Mossbauer factor fLM , a broadening ∆Γ of the lines, and a noticeable

decrease of the quadrupole splitting ~Ω. The temperature dependences of these three

parameters are shown in Fig. 6.1 for Fe2+ in glycerol [AM72, NFP91] and for ferrocene

in dibutyl phthalate [RZF76].

The Lamb-Mossbauer factor describes the fast dynamics on the time scale up to a few

ns (in the energy scale down to 200-1000 neV). In solids it is usually defined by lattice

vibrations which occur with characteristic times in the fs-ps time region . In harmonic

approximation and at temperatures above the Debye temperature the Lamb-Mossbauer

6.1. Previous studies of the LGT by Mossbauer spectroscopy 73

factor can be approximated by

fLM ' exp(−aT ) (6.1)

where the coefficient a is defined by the lattice vibrations. This approximation is also

valid for the glassy state of the glass former as seen in the figure. However, around and

above Tg a futher decrease of the Lamb-Mossbauer factor occurs which means a strong

deviation of ln fLM from the linear temperature dependence. Whereas such a behavior

was always observed in the studies of LGT by MS, a fundamental explanation has been

given only by the MCT. This theory predicts the existence of a relaxational process on

the picosecond time scale, the fast β relaxation, which leads to the anharmonic decrease

of fLM .

Another parameter which was studied in MS is the line broadening ∆Γ. It appears

due to relaxation on the ns time scale and is proportional to the relaxation rate λNFS.

The data sets which are presented in Fig. 6.1 show two common features: log ∆Γ is in-

verse proportional to T at low temperatures (Arrhenius relation) and deviates from this

dependence at high temperatures. Abras and Mullen [AM72] explain the turn of the

temperature dependence of ∆Γ by assuming that diffusion consists of two distinct pro-

cesses which additively contribute to the line broadening. The main contribution at low

temperatures is a thermally activated diffusion process which is described by an Arrhe-

nius temperature dependence. With an increase of temperature the second contribution

becomes important which is due to liquid-like diffusion and is proportional to viscosity.

However, both processes were assumed to be due to spatial relaxation. The contribution

of spin dynamics to the line broadening was ignored. The influence of spin relaxation has

been studied by Ruby et al [RZF76], who tried to separate spin and spatial relaxation

contributions to the line broadening. However, this was impossible in the frame of MS.

For this reason, it was either assumed [LRK77] or vaguely concluded [RZF76] that only

diffusion (spatial relaxation) gives rise to the line broadening ∆Γ.

In several works [GHL+98, NFP91, CHK+96] it was obtained that the line shape of

the Mossbauer spectra can not be described by a simple exponential diffusion law, but

requires a distribution of exponential relaxators. Kohlrausch-like relaxation functions

were used and characteristic stretching parameters were obtained.

The last parameter which we consider here is the value of the hyperfine field experi-

enced by the nuclear spin. For the presented studies there is the quadrupole interaction

between the nucleus and its surroundings inside a molecule. The temperature dependence

of the quadrupole splitting ~Ω is shown in Fig. 6.1 (middle). Whereas in the glassy state

a slight gradual decrease of ~Ω is observed, a rapid falloff of the quadrupole splitting

occurs during the glass-to-liquid transtion, in the temperature range where also the two

other parameters exhibit a deviation from the low temperature behaviour. This supports


the idea that the falloff in ~Ω is connected with the LGT and has a dynamical origin,

as was pointed out in [AM72, RZF76]. However, a clear explanation of this phenomenon

had not been given.

The discussion given above leads us to several open questions:

• How strong is the contribution of the spin relaxation to the line broadening? Can

one neglect this contribution?

• What is the origin of the change of the temperature dependence of the line broad-

ening?

• How one can explain the falloff of the quadrupole splitting in the region of the LGT?

Nuclear resonant scattering of SR allows us to answer these questions. The combi-

nation of NFS and SRPAC applied in parallel to investigate dynamics of a glass-forming

liquid extends the studied temperature range and allows one to separate spin and spatial

dynamics. The last feature leads to a separation of rotational and translational motions

in case of suitable probe molecules.

6.2 The sample

As a model substance we have chosen the molecular glass former dibutyl phthalate (DBP)

doped by 5% (mol) of ferrocene (FC) enriched by 57Fe. This system, (DBP-FC) has been

studied by MS [RZF76] (see Fig. 6.1) and by NFS [MFW+97, SFA+02].

Dibutyl phthalate (DBP) is a molecular glass former of medium fragility (D = 11)

with a glass transition temperature Tg = 179 K [BNAP93] and a melting temperature

Tm = 238 K [BGCF02]. The characteristic volume per molecule is VDBP = 441 A3 as

obtained from density. The structural formula is C6H4[COO(CH2)3CH3]2. A schematic

representation of the molecular structure is shown in Fig. 6.2 (left side).

Ferrocene (FC) or biscyclopentadienyl iron, Fe(C5H5)2, is a sandwich compound in

which the iron atom is located between two five-membered aromatic hydrocarbon rings.

The characteristic volume is 196 A3 as obtained from density. A schematic representation

of the molecular structure is shown in Fig. 6.2 (right side). X-ray analysis of crystals

of FC [DOR56] gives that the Fe-C distance is 2.0 A and the C-C distances around the

ring are 1.4 A. One can estimate that the Fe-ring distance is 1.6 A. As investigated by

EXAFS [KF01] dissolving FC in DBP does not change the Fe-C distance.

The electric field gradient is formed at the iron nucleus by π-bonding ligands and

directed perpendicular to the planes of the rings. The EFG produces a relatively large

quadrupole splitting, which for ferrocene powder is equal to ∼ 2.38 mm/s (24.5Γ0) at

80 K [GH68].

6.2. The sample 75

F eOCH

Figure 6.2: Schematic representation of a molecule of dibutyl phthalate (left side) and

ferrocene (right side).

Usually the spatial motions of an atom are separated into three contributions: the

intermolecular vibrations, the molecular rotation and the translational, center-of-mass

molecular motion. The first contribution occurs with a characteristic time much faster

than the typical relaxation time. The advantage of the FC molecule is that the iron atom

is in the center of mass. Therefore, its position does not change when the molecule rotates.

The spatial dynamics of the iron atom unambiguously corresponds to the translational,

center-of-mass molecular dynamics. At the same time, the direction of the nuclear spin on

the iron site is defined by the structure of the FC molecule. The rotation of this molecule

around axes different than ring-Fe-ring axis is seen via the nuclear spin dynamics. Thus,

the spatial dynamics of the iron atom in the FC molecule corresponds to translational

molecular dynamics and the spin dynamics of the iron atom corresponds to molecular

rotation.

Ferrocene enriched to 95% in 57Fe was prepared as follows. In a first step, FeCl3 was

obtained in 81% yield by the reaction of an enriched iron sheet with a Cl2 gas stream

in a quartz tube. After dissolving in diethyl ether, a solution containing a stochiometric

excess of freshly prepared cyclopentadienyl lithium (LiC5H5) in tetrahydrofuran was added

under a dry nitrogen atmosphere. Working up the product after stirring overnight and

subsequent sublimation gave an overall yield of 71% pure 57Fe(C5H5)2, which was dissolved

in dibutyl phthalate with purity 98% obtained from Sigma to give a 5% (mol) solution.

Such a concentration corresponds to 1 FC molecule per 19 molecules of DBP. The obtained

solution was inserted into a copper holder with size 10 × 4 × 2 mm3 sealed with Kapton

windows. The temperature measurements were performed using a closed-cycle cryostat

having a stability better than ±0.5 K. Temperatures below 160 K were achieved by fast

cooling to reach the glassy state. All measurements were done during heating sequences.

The caloric glass transition temperature of DBP-FC Tg = 178 ± 1 was determined by


Figure 6.3: Time evolution of the SRPAC intensity measured in the 16 bunch mode. The

solid line is the fit by the SCM. The dashed line denotes the natural exponential decay

with lifetime of 141.1 ns.

differential scanning calorimetry (with a heating rate of 10 K/min) in [MFW+97]. The

value of Tg coincides with the one for pure DBP.

6.3 Experiment by SRPAC

The measurements have been carried out at the European Synchrotron Radiation Facility,

Nuclear Resonance beamline ID18.

The first measurements have been performed in the 16 bunch mode (time window

∼ 171 ns). The temperature range was 180-250 K, and a reference measurement was

performed at 300 K. The setup of the experiment is presented in Section 5.1. The energy

bandwidth of the high resolution monochromator was 150 meV covering the entire range

of elastic and inelastic nuclear excitations. The detector (area 5x5 mm2) was placed in

90o scattering geometry, at 30 mm below the sample. The countrate was 1-2 Hz. Typical

time spectra are shown in Fig. 6.3. At 200 K the time evolution is characterized by a

natural decay modulated by a pronounced quantum beat (QB) which corresponds to the

6.3. Experiment by SRPAC 77

quadrupole splitting of ferrocene. The beat exhibits only a weak damping, indicating

slow relaxation. At 247 K the QB is largely overdamped due to the rotation of the FC

molecules. At 300 K the slow approach of the natural decay (dashed line) is character-

istic of fast relaxation. In brief those spectra give an overview on SRPAC influenced by

relaxation.

In order to make a systematic study of relaxation the experimental setup was modified

in three important points. In order to increase the countrate, a larger detector (area

10x10 mm2) was placed in 90o scattering geometry at 10 mm below the sample, yielding a

gain in the countrate by a factor of ∼36. A slight decrease of the contrast due to angular

average had to be tolerated. In order to reduce the load of the prompt radiation on the

detector, the bandwidth of the incident beam was reduced to 6 meV. In order to increase

the sensitivity, the experiment was performed in the single bunch mode (time window

∼2.8 µs), allowing for a large observation window. This permits to push both limits of

the dynamic range: the lower limit of slow relaxation, where a very weak damping of

the quantum beat modulation has to be observed, and the upper limit of fast relaxation,

where a very slow approach of the decay to the natural decay has to be followed. Both

effects are better seen over a large time window. Using this setup measurements have

been performed in the temperature range from 160 K (20 K below Tg) where DBP-FC is

in the glassy state, up to 330 K (90 K above melting temperature) where DBP-FC is in

the normal liquid state. The entire range from glass to normal liquid state was covered

by the measurements. Additionally, a reference measurement was performed at 100 K.

The obtained SRPAC intensities are shown in Fig. 6.4 for several temperatures. The

time evolutions are different at low and high temperatures corresponding to the static case

and to the slow and the fast relaxation regimes. The anisotropy A22G22(t), which was

extracted from the intensity according to eq. (5.1) is shown in Fig. 6.5 for the temperature

range 160 − 231 K and in Fig. 6.6 for the temperature range 240 − 330 K.

Below 200 K the time evolution of the anisotropy is described by a QB slightly damped

in time. The baseline of the beat is shifted up from zero due to the constant contribution

of the hard core (compare eq. 4.20). Additionally, at early times SRPAC is overshadowed

by the contribution of NFS-Rayleigh scattering (see section 5.2.2). This contribution

decays faster than the natural decay. In the anisotropy it is seen as a downward deviation

of the baseline at early times. The probability of NFS-Rayleigh scattering is proportional

to f 2LM . Therefore, this contribution becomes smaller with increasing temperature and

vanishes above 195 K. Below this temperature the data sets were fitted only in the time

windows above the times marked by arrows in Fig. 6.5.

At 200-235 K the anisotropy is described by a sum of the QB term and the hard core

term both damped in time, so that the anisotropy approaches zero at large times. The

envelope of the QB corresponds to the rotational correlation function, which is seen here


Figure 6.4: Time evolution of the SRPAC intensity for temperatures (from top to bottom):

100, 180, 211, 231, 263 and 296 K. The solid line is the fit by the SCM.

directly in time. The ratio between the decays of the QB and of the hard core is defined

by the model of rotation, i.e. by the characteristic jump angle. At 231 K the anisotropy

becomes overdamped, so that no QB is seen anymore. A fast decay from the initial value

to zero describes the anisotropy here.

The time evolution of the anisotropy in the fast relaxation regime (T > 235 K) is

characterized by an exponential decay from the initial value, which is A22, to zero at

infinite time. This decay is fast for T = 242 K and becomes slower with increasing

temperature. The decay rate here is proportional to the mean relaxation time. Therefore,

only the integral of the correlation function influences the anisotropy in this regime.

In order to extract information about dynamics from SRPAC data one has to find

the value of the anisotropy coefficient A22. Due to the fixed experimental setup and

therefore fixed geometry of the experiment, this coefficient has to be the same for all

temperatures. It is not reasonable to fit it as a free parameter for each data set. The

theoretical calculation of A22 as a function of distance d between the detector and the

sample, which was done in Section 5.2.1, gives A22 = 0.38 ± 0.015 for d = 10 ± 1 mm


Figure 6.5: Time evolution of the anisotropy for various temperatures in slow relaxation

regime. The solid line denotes the fit according to SCM. The arrows show beginning of

the fitted time window.

and a 10 × 10 mm2 detector. In order to check this value we fit the anisotropy in the

fast relaxation regime. This choice of the temperature range is explained as follows. In

the slow relaxation regime the time evolution of the anisotropy follows the time evolution

of the rotational correlation function, which is unknown in general. Therefore, the value

of the initial anisotropy A22 depends on the choice of this function. Also, the additional

contribution produced by NFS-Rayleigh scattering complicates the extraction of A22 from

the data. On the other hand, in the fast relaxation regime the anisotropy is well described

by the exponential decay independent of the correlation function. The obtained values

of A22 are shown in Fig. 6.7. One can see that the experimental results coincide with

theoretical calculations within the limits of error. This allows us to fix A22 = 0.38 for all

measurements.

In order to fit the SRPAC intensity I(t) in the slow relaxation regime we used numerical

calculations of the perturbation factor according to Appendix D. This model assumes an

exponential shape of the correlation function and includes three independent parameters:

the quadrupole splitting ~Ω which describes the frequency of the quantum beat, the


Figure 6.6: Time evolution of the anisotropy for various temperatures in the fast relaxation

regime. The solid line denotes the fit by SCM.

relaxation rate λ2 which describes the characteristic decay rate, and the parameter q

which enters into the ratio of decay rates of the QB and of the hard core and which

describes the model of rotation.

An attempt to use all three parameters as independent parameters of the fit procedure

was not successful. Whereas the decay rate of the quantum beat can be obtained from the

fit with reasonable error, the decay rate of the hard core practically can not be obtained

with this statistics of the data. In order to overcome this problem we have to introduce

the model of rotation a priory. We assume that molecular rotation occurs according to

the strong collision model (SCM). There is a good reason to take this model. Consider

the dependence of the relaxation rate λ2 for different values of q on the decay rate po of

the QB, which can be unambiguously obtained from the fit. According to Fig. 4.6 the

possible values of q are restricted to the range 0.4 < q < 3.33. This restriction leads to

(see eq. (4.25))

0.54 < po/λ2 < 0.7 (6.2)

or

0.86po

0.6< λ2 < 1.11

po

0.6(6.3)

As result we obtain that the relaxation rate λ2 is situated in the ±15% region around


Figure 6.7: The anisotropy coefficient A22 obtained from the fit of the SRPAC data. The

hatched area denotes the theoretical value A22 = 0.38 ± 0.015.

the unambiguously obtained value of po/0.6 which is the decay rate in the SCM (q = 1)

according to eq. (4.20). Thus, one can write λ2 ' λSCM2 (1 ± 0.15) which means that

independent of the model of rotation, the relaxation rate λ2 is situated in the ±15%

region around λSCM2 . This conclusion allow us to work in the frame of the SCM. For this

model an analytical expression for the perturbation factor exists which is described in

Appendix E.

The approach presented above is valid only in the slow relaxation regime and applies

only to the relaxation rate λ2. In order to estimate in general the sensitivity of the

relaxation rate and the quadrupole splitting on the choice of the model we have treated

the experimental data in addition using the RDM (q = 20/6). This treatment uses the

numerical procedure of Appendix D. The comparison of the relaxation rate and the

quadrupole splitting for the two different models of rotation gives their possible range.

Fig. 6.8 shows the temperature dependence of the quadrupole splitting ~Ω obtained

in the SCM and the RDM and of the effective quadrupole splitting ~Ωe which is the

true frequency of the QB. The quadrupole splitting is almost constant up to 180 K and

decreases above. The three sets of data are identical up to 200 K and diverge above. It

is shown in the theoretical part of this work (see eq. (4.21)) that the effective quadrupole

splitting ~Ωe decreases due to relaxation as ∆Ωe ∝ −λ22. This fact explains the deviation

of ~Ωe from ~Ω obtained in the SCM. On the other hand, the increase of ~Ω obtained in

the RDM is artificial, probably, due to the wrong choice of the model of rotation. The

coincidence of all data sets below 200 K means that the influence of the relaxation is

vanishing below this temperature and that the decrease of ~Ω in the region 180-200 K

can not be explained by the influence of slow rotational relaxation. We attribute this

decrease to fast librations of the ferrocene molecules in a restricted angular range which


Figure 6.8: Temperature dependence of the effective quadrupole splitting ~Ωe (¤) and of

the quadrupole splitting ~Ω obtained by SRPAC using the SCM (•) and the RDM (4).

The solid line denotes the extrapolation to high temperatures.

lead to a pre-average of the quadrupole splitting as it is shown in Section 4.4.4. The

corresponding interpretation of the data will be given later.

The quadrupole splitting can be obtained as an independent parameter only up to

221 K. At higher temperatures λ2 and Ω form one parameter, Ω2/λ2 and can not be fitted

independently. Therefore, in order to extract λ2 one has to introduce some information

about Ω at the temperature range above 225 K. We assume a linear extrapolation of ~Ω

as

~Ω = 25.85 · (1 − T/3047) (6.4)

This dependence is shown in Fig. 6.8 by a straight line. It reproduces the temperature de-

pendence of ~Ω obtained in the SCM up to 221 K. The real dependence of the quadrupole

splitting on temperature can deviate from this linear approximation. However, the possi-

ble uncertainties are not significant, because the variation of Ω with temperature is much

smaller than that of λ2.

Fig. 6.9a shows the temperature dependence of the relaxation rate λ2 obtained using

the SCM and the RDM. The relaxation rate increases from ∼ 0.2/τ0, at 160 K up to

5.6 · 103/τ0 at 328 K. As a whole, reliable data are obtained within a frequency range

covering 5 decades. The data sets obtained using the two different models show similar

behaviour. At low temperatures they are shifted by a scaling factor of 1.1 − 1.2 which is

in agreement with our estimation (see eq. (6.3) and discussion). Above 230 K data sets

nicely coincide corresponding to the independence of the relaxation rate on the model of

rotation in the fast relaxation regime.


160 200 240 280 320

10-2

10-1

100

b

Dam

ping

rat

e [

Ω ]

Temperature T [ K ]

100

101

102

103

104

a

Rel

axat

ion

rate

λ2

[ τ-1 0 ]

Figure 6.9: a) Temperature dependence of the relaxation rate λ2 obtained by SRPAC

using the SCM (•) and the RDM (4). b) Temperature dependence of the damping rate

of the anisotropy in slow relaxation regime (¥) and in the fast relaxation regime ().

In order to see a separation of the whole temperature range into slow and fast relax-

ation regimes we plot the characteristic damping rate of the anisotropy in Fig. 6.9b. For

low temperatures only the damping of the QB contribution in the anisotropy is shown

which has the rate 0.6λ2 (see eq. (4.20)). At high temperatures the damping rate of the

anisotropy is given by 0.8Ω2/λ2 (see eq. (4.26)). One can see that the damping increases

with temperature in the low temperature region and decreases in the high temperature

region. The two curves cross at 230-240 K and in this region Ω ∼ λ2. The data sets at

231 and 242 K have maximum damping rate where the anisotropy falls to zero right from

early times on.

Fig. 6.9b allows us to define the frequency range which can be investigated by SRPAC.

The minimum damping rate is defined by the experimental time window and by the

life time τ0 of the excited state. In the present measurements the minimum damping

rate is ∼ 0.1/τ0. This corresponds to λ2 ∼ 0.16/τ0 as a minimum relaxation rate and

to λ2 ∼ 8Ω2τ0 as a maximum relaxation rate. The separation into the slow and fast

relaxation regimes occurs around a maximum damping rate, where λ2 ∼ Ω. In the present

measurements the temperature range around 230-240 K corresponds to the intermediate


relaxation regime. Accordingly, the slow relaxation regime extends up to ∼230 K and the

fast relaxation regime starts at ∼240 K.

6.4 Experiment by NFS

The measurements at low temperatures have been performed in the hybrid bunch mode

(time window ∼ 500 ns). In these measurements NFS alone has been observed in the

temperature range 25-190 K. The used sample had a thickness of 12 mm. Typical time

spectra are shown in Fig. 6.10.

A second set of NFS measurements has been performed in parallel with SRPAC mea-

surements in the temperature range 160-211 K. While SRPAC measurements were con-

tinued above 211 K, the countrate of the NFS signal becomes too low due to a vanishing

Lamb-Mossbauer factor. The measured time spectra are shown in Fig. 6.11. The detec-

tion of the delayed signal starts at ∼20 ns in both sets of measurements due to overload

of the detector.

The time evolution of the NFS intensity is defined by a combination of the exponential

decay, the QB due to the quadrupole splitting of the excited state and the dynamical

beat (DB) due to multiple scattering.

The QB is the most pronounced feature of the data sets. The period of the QB is

∼36 ns and is almost constant with temperature. At low temperatures the QB is strongly

perturbed by the interaction with the DB.

The DB is shown by arrows in Fig. 6.10. It follows the Bessel-like dependence in

time. Periods of the DB increase with time. A characteristic frequency is proportional to

the effective thickness ξ of the sample and to the Lamb-Mossbauer factor fLM . At low

temperature the value of ξfLM is so large that six DB minima are seen in the experimental

time window up to 500 ns. The Lamb-Mossbauer factor decreases with temperature, which

leads to a stretching of the DB in time. Whereas the first minimum is around 20 ns at

50 K, it shifts to 45 ns at 124 K, to 70 ns at 150 K and to 180 ns at 177 K. One can see

that the stretching increases drastically between 150 and 180 K, which is connected with

a strong decrease of fLM in this temperature region. The second set of measurements

uses a sample with 1.2 times smaller thickness which is seen in the comparison of the time

spectra at 177 K (Fig. 6.10) and at 180 K (Fig. 6.11). While fLM is almost the same,

the position of the first DB minimum is different. This is due to the different value of ξ.

Above 190 K no DB minima are observed in the experimental spectra. At these elevated

temperatures the effect of the DB reduces to an additional exponential decay with a

decay rate proportional to ξfLM according to the quasi-kinematical approximation (see

eq. (3.11)).

6.4. Experiment by NFS 85

Figure 6.10: Time evolution of the NFS intensity at low temperatures measured in the

hybrid bunch mode. The solid line denotes the fit by the model (see text). The arrows

show minima of the DB.

The relaxation appears as an additional damping of the intensity. Above 200 K a

non-exponential (stretched) decay of the NFS intensity becomes visible.

The treatment of the data measured below and above 160 K has been performed in a

different way. Below 160 K relaxation is absent and the numerical calculation of the NFS

intensity was used according to eqs. (3.3)-(3.5). The free parameters were the incoming

intensity I0, the effective thickness parameter ξfLM and the quadrupole splitting ~Ω.

In order to obtain ξ and fLM separately, the obtained values of ξfLM were compared

with fLM obtained by nuclear inelastic scattering (NIS) at 25, 50, 100, 150 K on the same

sample [Chu03]. As a result, the effective thickness ξ = 87 is obtained for the first sample.

Comparison of ξfLM of the first and second samples gives ξ = 73 for the second sample

which corresponds to the ratio of thicknesses of the samples.

The treatment of the data measured above 160 K meets several problems which have

to be solved. The first problem is the shape of the relaxation function which has to be


Figure 6.11: Time evolution of the NFS intensity at high temperatures measured in the

single bunch mode in parallel with SRPAC. The solid line denotes the fit by the model

(see text).

introduced into the theoretical model. The experimental data manifest a non-exponential

character of the relaxation above 200 K. On the other hand, the developed general theo-

retical model for SRPAC and NFS implies an exponential relaxation. In this section we

limit the treatment of the data to an exponential shape of the relaxation function taking

into account that the same exponential relaxation was used to treat the SRPAC data and

that both NFS and SRPAC were measured in the same time region. This fact allows us

to compare results obtained in both methods. Later, we consider the treatment of the

data by a Kohlrausch relaxation function.

The theoretical expression describing the NFS intensity in the presence of spatial and

spin dynamics is developed in Section 4.5. The free parameters here were the incoming

intensity I0, the effective thickness parameter ξfLM , the relaxation rate λNFS and the

effective quadrupole splitting ~Ωe. The relaxation rate λNFS is the sum of λ2/2 which de-


Figure 6.12: Temperature dependence of the countrate R.

scribes molecular rotation and λt which describes translational, center-of-mass molecular

motion. The effective quadrupole splitting ~Ωe differs from the quadrupole splitting due

to the influence of rotational relaxation.

Above 190 K the DB minima are not seen in the experimental time window and an

evaluation of ξfLM from the DB structure is not possible. Moreover, the damping of the

intensity is formed by a sum of ξfLM/2τ0 and 2λNFS. In order to separate these two terms

we use another way to obtain ξfLM . The scattered intensity at time zero is proportional

to ξ2f 2LM according to eq. (3.10). Then, by scaling of the obtained data sets to the same

incoming intensity one can obtain the dependence of ξfLM on temperature from the value

of the intensity at time zero. However, the experimental setup does not provide a long

time stability of the incoming radiation level. Therefore, it is very difficult to estimate

the intensity incident on the sample during the entire time of measurement (from 0.5 up

to 8 hours).

In order to overcome this problem we measure the integrated NFS intensity during

a short period of time. We call this value countrate R. During this time the incoming

beam is stable and its intensity is measured. The countrate is scaled to the unit of time

and to the unit of the incoming intensity. The countrate obtained this way is shown in

Fig. 6.12 for temperatures above 180 K. Theoretically, the integral of the NFS intensity

in the quasi-kinematical approximation gives

R ∝∫

I(t)dt ∝ ξ2f 2LM

1 + ξfLM/2 + 2λNFSτ0(6.5)

The denominator here is the decay rate of the intensity and can be extracted from the time

spectra. The proportionality coefficient can be found at 180 K where ξfLM is extracted

straightforwardly from the DB structure and where the quasi-kinematical approximation


starts to be valid. The value of ξfLM obtained from the countrate, was used to separate

the decay rate of the intensity and to get the relaxation rate λNFS.

The eq. (6.5) which describes the integral of the NFS intensity does not take into

account the QB and the cutoff of the experimental time window below 20 ns. However,

both factors lead only to a small correction of the expression. A more important correction

can appear if we consider relaxation governed by the Kohlrausch function. Then, the fast

decay at early times that is a part of the structural relaxation process is not visible in

the experimental time window. The approximation of the visible part of relaxation by

exponential function leads to an artificial decrease of the Lamb-Mossbauer factor.

6.5 Results and discussion

In the present measurements nuclear resonant scattering has been applied to study the

dynamics of the glass former DBP-FC. In particular, NFS has been studied up to 211 K

and SRPAC - up to 330 K. Below 221 K four parameters are obtained: the relaxation

rate λ2 from SRPAC, the relaxation rate λNFS from NFS, the quadrupole splitting ~Ω

from SRPAC and NFS and the Lamb-Mossbauer factor fLM from NFS. The relaxation

rates are connected with the slow dynamics which is seen directly in the experimental

time window. ~Ω and fLM are influenced by the fast dynamics restricted in space, like

vibrations, rattling and librations.

In the 220-330 K temperature range only one parameter is obtained - the relaxation

rate λ2. It gives information about the mean relaxation time for the rotational correlation

function C2(t) (see Section 4.4.2).

6.5.1 Fast dynamics

Fig. 6.13 shows the temperature dependence of the Lamb-Mossbauer factor. The data

can be described phenomenologically by a multiplication of two functions: the first one

decreases exponentially with temperature; the second function equals unity up to 100 K

and decreases faster than exponential above. Such factorization has a physical explana-

tion. The Lamb-Mossbauer factor corresponds to the decay of the self-correlation function

due to the fast molecular motions localized in space. Two processes define these motions

in glasses: vibration and fast localized relaxation. We assume the independence of these

processes. Therefore, the Lamb-Mossbauer factor can be factorized as

fLM = fh · fr (6.6)

where fh corresponds to the vibrations and fr to the relaxation.

6.5. Results and discussion 89

Figure 6.13: Temperature dependence of the Lamb-Mossbauer factor fLM obtained by

NFS (•,N), NIS (O) and from countrate R (♦). The solid line denotes the vibrational

contribution fh to the Lamb-Mossbauer factor. Inset: temperature dependence of the

relaxational contribution fr to the Lamb-Mossbauer factor obtained by fr = fLM/fh.

The solid line denotes the fit according to the square root dependence of eq. (6.8).

In order to obtain the temperature dependence of fh we consider the results of NIS

on the same sample [Chu03]. This method gives direct access to the density of phonon

states. The partial phonon density of states (DOS) g(E) of the iron nuclei in DBP-FC

is shown in Fig. 6.14a as measured at 50 K. Above 15 meV, it exhibits three narrow

peaks. These are the eigen modes of the ferrocene molecule, which involve displacements

of the central iron atom. The vibrational states below 15 meV describe displacements

of the rigid FC probe driven by the correlated motions in DBP-FC. The reduced DOS

g(E)/E2 of the correlated motions is shown in Fig. 6.14b for various temperatures. This

function remains the same up to 100 K. Above, it enhances at low energy with increasing

temperature. We identify these additional modes with fast localized relaxation via a

rattling of the molecules in cages of their neighbors, which becomes more pronounced

with temperature. The identity of the reduced DOS below 100 K allows us to conclude

that g(E) here describes the pure vibrational density of states. This function is used to

calculate fh by [SS60]

fh = exp

(

−ER

∫ ∞

0

g(E)

E· 1 + exp(−E/kBT )

1 − exp(−E/kBT )dE

)

(6.7)

where ER is the recoil energy and kB is the Boltzmann constant. The obtained tempera-


Figure 6.14: a)The partial density of states g(E) of the iron nuclei in DBP-FC at 50 K

obtained by NIS [Chu03]. b) The reduced partial density of states g(E)/E2 in DBP-FC

for various temperatures obtained by NIS [Chu03].

ture dependence of fh is shown by the solid line in Fig. 6.13. ln fh exhibits an almost linear

dependence on temperature above ∼50 K which can be easily derived from eq. (6.7). Using

fLM and fh we obtain the relaxational factor fr which is shown in the inset of Fig. 6.13 as

a function of temperature. In the MCT the fast localized relaxation, i.e. fast β relaxation

is introduced and fr is expressed by a square-root dependence (see eq. (3.8))

fr = f0 + hr

√

1 − T/Tc (6.8)

where Tc is the crossover temperature where the mechanism of relaxation is changing

from thermal activation to liquid-like motion. The fit of the data by this function gives

f0 = 0, hr = 1.3 and Tc = 195 K. The fit is shown by the solid line in the inset of

Fig. 6.13. The curve qualitatively describes the decrease of fr in the temperature range

from 80 up to 195 K. According to MCT fr above Tc is described by a weak dependence

on temperature (see eq. (2.8)). In principle, a similar behaviour of the data is seen above

195 K. This similarity, however, should be considered with care due to the difficulties in

evaluation discussed at the end of the Section 6.4. Also, one should remember that the

approximation (6.8) is valid only near Tc.


Figure 6.15: Temperature dependence of the quadrupole splitting ~Ω obtained by NFS (♦)

and SRPAC using the SCM (•). Inset: temperature dependence of the rotational factor

frot obtained by frot = (~Ω/24.68Γ0)2. The solid line denotes the fit according to the

square root dependence of eq. (6.8).

Another interpretation of the Lamb-Mossbauer factor can be given using the mean-

square displacement < r2 > of the iron atom

fLM = exp(−k2 < r2 >). (6.9)

where the wave vector k = 7.3 A−1 corresponds to the resonant energy. Similar to

eq. (6.6) the mean square displacement can be presented as a sum of vibrational < r2 >h

and relaxational < r2 >r contributions. The vibrational contribution can be evaluated as

before. The temperature dependence of the relaxational contribution is shown in Fig. 6.16

in comparison with the mean square angular displacement.

The temperature evolution of the quadrupole splitting is shown in Fig. 6.15 as ob-

tained by SRPAC and NFS. The quadrupole splitting remains constant up to ∼100 K

and decreases above. It was already shown that the slow relaxational process cannot

explain this decrease. We rather attribute it to fast librations of the FC molecules in a re-

stricted angular range which leads to an averaging of the electric field gradient and of the

quadrupole splitting over angle. It was shown in Section 4.4.4 that these fast librations

can be described by the rotational factor frot which is similar to the Lamb-Mossbauer

factor but in the angular space. According to eq. (4.32)

Ω

Ω0

=√

frot ' 1 − 3

4< α2 > (6.10)


Figure 6.16: Temperature dependences of the relaxational part < r2 >r of the mean

square displacement (♦) obtained from the Lamb-Mossbauer factor and of the mean

square angular displacement < α2 > (•) obtained from the quadrupole splitting. The

left scale corresponds to < α2 > and the right scale corresponds to the < r2 >r. The

scaling factor is < r2 >r=< α2 > ·2.56A2.

where < α2 > is the mean square angular deviation and the second equality is obtained

in the approximation of small < α2 >. The quadrupole splitting exhibits a constant

dependence below 100 K, which leads to frot = 1 at those temperatures. Using ~Ω0 =

24.68 we obtain the rotational factor which is shown in the inset of Fig. 6.15. This factor

can also be fitted by the square-root dependence of eq. (6.8). The solid line in the inset

of the figure shows this fit. The obtained parameters are f0 = 0.95, hr = 0.08 and

Tc = 206 K. Again, the data show a weak temperature dependence above 206 K which

corresponds to the MCT prediction. However, the uncertainty of the data increases

drastically above this temperature. Also, ~Ω here strongly depends on the chosen model

of rotation as was shown in Fig. 6.8.

The treatment of both fLM and frot leads to the conclusion that the crossover tem-

perature Tc is in the region of 195-206 K. Also we can use the result of the treatment

of NFS measurements on DBP-FC using the Kohlrausch relaxation function [SFA+02].

Here, Tc = 202 K has been obtained from the temperature dependence of fLM . Combining

the results we obtain Tc = 201 ± 6 K as a good estimate for the crossover temperature.

The rattling contribution to the mean square displacement < r2 >r can be compared

with the mean square angular displacement < α2 > which is obtained from frot according

to eq. 6.10. The temperature dependences of both parameters are shown in Fig. 6.16. The


two data sets show similar temperature dependences and mainly coincide when scaled

appropriately. This striking similarity proves that the fast angular librations of the FC

molecule and the fast spatial displacement of its center of mass are tightly coupled and

caused by the same process, the fast β relaxation. The scaling coefficient is

√

< r2 >r = 1.6A ·√< α2 >. (6.11)

This coefficient corresponds to the distance between the iron atom and the hydrocarbon

rings. A similar coincidence of the temperature behaviour has been observed in [SFS92],

where the mean square displacement and the mean square angular displacement of a

side group of a molecule have been investigated. The authors identified the coincidence

with rotation of the side group around a stiff molecular center of mass. Respectively, the

scaling coefficient is the distance between the center of mass and the side group. However,

this simple explanation does not work in our case, since we observe formally independent

processes of center-of-mass translation and rotation around center of mass.

6.5.2 Slow dynamics

Fig. 6.17 shows the rotational relaxation rate λ2 obtained from SRPAC as a function of

inverse temperature 1000/T . One can see that λ2 exhibits a different behaviour above and

below 210 K. Whereas a linear Arrhenius dependence is observed below this temperature,

a non-Arrhenius viscosity-like behaviour is seen above. According to the MCT we explain

this change of behaviour by different mechanisms of relaxation. The thermally activated

hopping process drives relaxations at low temperatures and the coupling between modes of

motion in the ensemble of molecules defines the structural relaxation at high temperatures.

Therefore, we fitted the data by different functions at low and high temperatures. The

Arrhenius law was used to fit λ2 below 210 K

λ2 = λ02 exp(∆E/kBT ) (6.12)

The fit is shown by the solid line in Fig. 6.17 and gives λ02 = 2.4 · 1011 Hz and ∆E =

2130K · kB = 17.7 kJ·mol−1.

The data at high temperatures were fitted by the MCT power law (see eq. (2.3))

λ2 = λ02 (T/Tc − 1)γ (6.13)

The temperature dependence of the relaxation rate here is defined by two parameters: γ

and the crossover temperature Tc. They hardly can be fitted together. The fits with Tc

chosen as 180, 201 and 220 K are shown in the inset of Fig. 6.17. The obtained γ equals

5.4, 4.4 and 3.3, respectively. One can see that the best fit corresponds to Tc = 180 K.

However, this result is physically incorrect since Tc must be well above Tg. From the two


Figure 6.17: Inverse temperature dependence of the rotational relaxation rate λ2 obtained

from SRPAC (•), the relaxational rate 2λNFS obtained from NFS (♦) and the line broad-

ening ∆Γ obtained from MS (¤) in [RZF76] . The solid lines denote fits by an Arrhenius

law above 1000/T = 4.8 and by a MCT power law below 1000/T = 4.4. Tc = 201 K was

chosen for the MCT power law. Inset: fit of λ2 by the MCT power law with Tc chosen as

180 K (dashed line), 201 K (solid line) and 220 K (dotted line).

other curves we hardly can choose the better one. In order to do so we use the information

about Tc which was acquired by the investigation of fast dynamics where Tc = 201 K has

been obtained. The solid line in Fig. 6.17 shows the MCT power law fit with Tc = 201 K,

γ = 4.4 and λ02 = 3.1 · 1011 Hz.

The fit by those two models allows us to divide the entire temperature region into

three parts: above ∼240 K, where supercooled liquid dynamics is observed, below 210 K

where thermally activated dynamics is observed and between 210 and ∼240 K where the

transition from one to the other dynamics occurs.

Additionally to λ2, we plot in Fig. 6.17 the relaxation rate 2λNFS obtained from NFS

and the line broadening ∆Γ obtained by MS on a similar sample in [RZF76]. The last value

is equivalent to 2λNFS so that the two data sets have to coincide which is indeed realized


Figure 6.18: Comparison of the rotational, λ2 (•) and translational, 2λt (♦, ¤) re-

laxation rates of the probe FC molecule obtained by nuclear resonant scattering (left

scale) to the dielectric spectroscopy data for pure DBP (right scale) from [DWN+90] ()and [DT87] (F). The solid lines denote the Arrhenius and MCT power dependences; they

are the same as in Fig. 6.17.

in our case. It is shown in Section 4.5 that the relaxation rate observed in NFS consists of

a rotational and a translational contribution as given by 2λNFS = λ2 +2λt where λ2 is the

same rotational relaxation rate as in SRPAC and λt is the decay rate of the exponential

van Hove self correlation function with scattering vector k = 7.3 A−1. Physically it means

that de-coherence of NFS in time is produced additively by two contributions: stochastic

molecular rotation which is seen via spin dynamics and translational relaxation which is

seen due to the collective character of NFS.

From a comparison of the NFS, MS and SRPAC data we see that below ∼195 K all data

sets coincide which corresponds to λ2 ' 2λNFS and, therefore, λt ' 0. More precisely, λt

is less than the limit of sensitivity of nuclear resonance scattering i.e., λt < 7 · 105 s−1.

Thus, below 195 K, the position of the FC molecule is frozen on the molecular length

scale and ns time scale. Above 195 K, however, 2λNFS begins to deviate from λ2 because


translational dynamics becomes comparable or even faster than rotational dynamics.

The pure translational relaxation rate 2λt can be derived and compared with λ2. It

is done in Fig. 6.18. In the same figure we compare our results for the dynamics of

the probe FC molecules in DBP to data on pure DBP available from dielectric spec-

troscopy (DS) [DT87, DWN+90]. The DS data were scaled by factor of 4 to match our

data at room temperature. At high temperatures the relaxation measured by DS consists

of a single branch, which coincides with our data for the rotational relaxation of the FC

probe molecule. At low temperatures the DS data split into two branches: structural

α relaxation and slow β relaxation. The slow β relaxation is followed by our data on

rotational molecular dynamics whereas structural α relaxation decreases in parallel with

our data on translational dynamics. The coincidence of the slow β relaxation branch of

DS and our data allows us to conclude that the rotation of the probe is driven by the

same process of slow β relaxation. On the other hand, the FC molecules remain in the

same space position on a sub-molecular length scale and on the time scale of slow β relax-

ation. We can conclude that, probably, the same absence of the translational motion is

realized for DBP molecules as well. As a result, slow β relaxation can be identified with a

certain mode of non-translational motion which decouples from the structural relaxation

below Tc. This mode of motion is molecular rotation in the case of the FC molecules and,

probably, for DBP molecules as well. Molecular rotation in the frozen glass structure

can explain the origin of slow β relaxation in the rigid molecular glass formers that was

discussed in Section 2.5.

We should mention here the result of [FGSF92] where a comparison of rotational dy-

namics measured by NMR and translational dynamics measured by various tracer tech-

niques has been performed for o-terphenyl. The authors obtained that the rotational

relaxation rate increases faster with temperature than the translational one. Whereas at

Tc both relaxation rates are comparable, at Tg, the translational relaxation is more than

2 orders of magnitude faster than rotational relaxation. This result looks opposite to that

obtained by us. However, whereas we investigate the time scale corresponding to slow β

relaxation, in [FGSF92] the time scale of structural α relaxation has been studied. Also,

nuclear forward scattering deals with the dynamics on a molecular length scale whereas

tracer techniques are sensitive to the translational motions on a macroscale.

In our investigation the rotational dynamics decouples from α relaxation. However,

we cannot say that there is no other branch of rotational relaxation which follows the α

branch. There are mainly two reasons why this branch is not seen. At first, the SRPAC

theory has been developed in the assumption of only one relaxation process. Secondly,

nuclear resonant scattering on 57Fe is not sensitive to the relaxation with frequency smaller

than 105 − 106 Hz.

A crossover temperature Tc = 220 − 230 K in DBP has been found in several stud-

6.6. Stretched exponential relaxation 97

ies [BGCF02, Ros90]. This value is significantly larger than our Tc = 201 K. However,

in [Ros90] Tc has been obtained from the fit of the viscosity by a MCT power equation. As

we saw (compare inset of Fig 6.17), without additional independent information this fit is

not precise and strongly depends on the chosen temperature region. In [BGCF02], where

DBP was studied by the heterodyne detected optical Kerr effect, the fit of the rotational

relaxation time has been performed by a MCT power equation. The power parameter γ

was fixed there via critical exponent of the fast β relaxation. Whereas qualitatively the

obtained curve describes the experimental data, a systematical deviation is seen, which

is, probably, due to a incorrect choice of γ and which leads to an artificial increase of Tc.

In general, we can conclude that Tc = 201 K obtained in our study from the dependence

of the Lamb-Mossbauer factor and of the quadrupole splitting on temperature at T < Tc

gives a lower limit of the estimation region of Tc while Tc = 227 K obtained in [BGCF02]

from dependence of the relaxation rate on temperature at T > Tc gives the upper limit

of this region.

The above interpretation of the results is restricted by the assumption of an expo-

nential character of relaxation. On the other hand, it is well known that wide frequency

distributions describe relaxation processes in glass formers. However, we use the fact that

both NFS and SRPAC measurements were done in the same experimental time window.

As result, the comparison of the NFS and SRPAC data is hardly sensitive to the choice of

the relaxation function. Moreover, in the fast relaxation regime, SRPAC is independent

on the relaxation function and gives the mean relaxation rate.

6.6 Stretched exponential relaxation

In the previous section the relaxation which appears in NFS was assumed to be expo-

nential. This allows us to factorize the relaxation into a translational and a rotational

contribution and, with the help of SRPAC, to find the rate of each contribution. On the

other hand, we can assume that the dynamics which is seen in NFS is described by the

generalized self-correlation function which includes both types of motion [GSV00]. This

function can be described by a Kohlrausch function (see eq. (3.21)). The two parameters

of this function are the relaxation rate λt and the stretching parameter β which is the

feature of non-exponentiality of this function. In this section we do not take care about

the type of motion which is seen in NFS, but will only try to extract information about

the parameter β. In order to do so it is helpful to make an additional data treatment.

The QB which is seen in the NFS time evolution occurs with a characteristic time

faster than the characteristic times of all other processes. Also, the QB and the relaxation

contribution to NFS intensity are independent. Therefore, in order to avoid additional


Figure 6.19: Data treatment procedure: original experimental data (¯), modified data

corrected to the average intensity (¨).

parameters in the fit procedure and to increase the statistics in the long time window the

integration of the time spectra over the QB oscillations can be done before fitting. The

integration was performed for each beat period and the time abscissa of the integrated

data were set to the positions of the maxima of the QB pattern. The resulting set of points

can be considered as a response of the ensemble of the nuclei with a single resonance and

the theory introduced in Section 3.4.3 can be applied for the treatment of a data.

The correctness of this procedure was checked numerically by the fit of the full and

the reduced set of the data measured at 180 K. The comparison of the results shows an

agreement within the limit of the experimental error. In Fig. 6.19 the original and the

modified data are shown for two temperatures. One can see that for T = 206 K the

reduced data look more informative in the time region after several τ0 than the original

one.

The left side of Fig. 6.20 shows the time evolution of the reduced NFS intensity for

the temperatures 180-206 K. The experimental data were fitted using the numerical cal-

culation of the NFS intensity for a single resonance and a Kohlrausch relaxation function

according to Appendix A. The accurate estimation of the stretching parameter β requires

the observation of the signal at least over two decades of time [GS92]. The experimental

time region is ∼ 1.5 decades (30 ÷ 800 ns) and an estimation of β as a free parameter of

the fit is difficult. In order to solve this problem, β was chosen as β = 1, 0.8, 0.6, 0.5, 0.4.

For each β the sum of least square residuals χ2 of the least-squares fit procedure was

calculated. χ2 as a function of β is shown in the right side of Fig 6.20. One can see that

6.6. Stretched exponential relaxation 99

Figure 6.20: Left side: time evolution of the reduced NFS intensity for various tempera-

tures. The lines show the fit with relaxation described by the Kohlrausch function with

β = 0.5 (solid line) and β = 1 (dashed line). Right side: the sum of least square residuals

of the fit χ2 as a function of parameter β.

χ2 monotonically decreases with β decreasing from 1 to 0.5. However, the dependence

of χ2 on β becomes less pronounced when β comes to ∼ 0.5. This can be explained as

follows.

Fig. 3.3 shows the distribution of the Debye (exponential) relaxators corresponding

to the Kohlrausch function. The distributions for β = 1, 0.9, 0.7 extend over ∼ 1 decade

of frequency (corresponding to one decade of time). Taking the part of the distribution

corresponding to one decade, it is possible to restore the full distribution. On the other

hand, the distribution for β = 0.3 can not be restored taking only one decade of frequency

range. The one decade part of the distribution near the center is almost constant and

hardly gives information about the tails. The value β ∼ 0.5 corresponds to the border

value. We can say that the observed experimental data can not be described by β > 0.5.

However, the experimental time window does not allow us to find a precise value of β.

Phenomenologically, one can work with the maximal possible β. This value was estimated


as β = 0.47 in [SFA+02] which roughly corresponds to the presented experimental data.

Chapter 7

Study of LGT dynamics in restricted

geometry by NFS

The application of NFS to study dynamics of a glass-forming liquid confined in pores is

presented in this chapter. After a brief introduction and a description of the sample the

experiment by NFS is described. The obtained results are compared with the results in

the bulk case in the last part.

7.1 Dynamics of viscous liquids in restricted geome-

try

Studying the relaxation dynamics of liquids and glass formers in restricted geometries has

attracted much interest recently [FZB00] for mainly two reasons. First, it is obviously

important to understand how fluid systems behave on interaction with surfaces, e.g. on a

catalyst, in a capillary or in a mesoscopic material, especially with respect to a multitude

of applications in chemical and biological sciences. Second, it is believed that confinement

may also help to address more fundamental questions about bulk dynamics. For example,

the existence of diverging timescales at the glass transition is now widely accepted [Got99],

whereas the question whether there is also an associated characteristic length scale remains

an unsolved question up to date. This question can be addressed by introducing the glass

former into a geometrically confined environment, in order to test qualitative changes in

the glass dynamics when the characteristic length exceeds the size of the confined space.

Here, it is mandatory to minimize strength and range of the interaction between the

glass and its surrounding matrix, so that purely geometrical constraints dominate the

dynamical behavior.

During the last 15 years many different techniques were applied to such systems, for a

recent review see [FZB00]. Surprisingly, the obtained results do not give a clear answer at

101

102 Chapter 7. Study of LGT dynamics in restricted geometry by NFS

all. For certain systems the caloric glass transition temperature in the confined system is

found to be higher than in the bulk system [SMRF94], whereas in other systems it seems

to be lower [JM91].

The experimental investigation of the dynamics of glass formers inserted into a matrix

which defines the restricted geometry is often complicated due to a strong background

signal from the matrix. The unique advantage of NFS is that this method is a background-

free method: only the iron-containing glass former is observed since only the resonant

nuclei contribute to the delayed signal. The matrix itself gives a signal at t = 0, which is

discriminated in the experiment. Another advantage of NFS is that small samples may

be used due to the high brilliance of SR.

7.2 The sample

The molecular glass former dibutyl phthalate (DBP) doped by 5% (mol) of ferrocene (FC)

enriched by 57Fe was used as a model substance. The study of this glass former in the bulk

case was presented in the previous chapter. The preparation of DBP-FC was described

in Section 6.2.

The nanoporous samples were prepared from commercially avaliable (Geltech Inc.,

Orlando, USA) SiO2 aerogel pellets prepared by a sol-gel process and having pore sizes

25, 50, 75 and 200 A. In order to optimize the countrate including the influence of pho-

toabsorbtion, pellets having 6 mm diameter and 2 mm thickness were ground down to

half-disks of 1.1-1.5 mm thickness.

The loading procedure was performed by T. Asthalter and A. Huwe and is described

in [ASF+01, ASG+97]. In order to avoid the chemical coupling between the glass former

and the wall, the pore walls were lubricated using hexamethyldisilazane before adding

DBP-FC.

7.3 Experiment by NFS

The measurements have been carried out at the ESRF, Nuclear Resonance beamline ID18.

The first measurements have been performed in the 16 bunch mode (time window ∼171 ns). Pellets with 25, 75 and 200 A pore sizes were studied in the temperature range

80-200 K. The usual setup for NFS was used. Typical time spectra are shown in Fig. 7.1

for 25 and 200 A pore sizes. The results of this study have been published in [ASF+01].

The time evolution of the intensity is characterized by an exponential decay modulated

by a pronounced QB due to the quadrupole splitting of the iron nucleus in the ferrocene


Figure 7.1: Time evolution of the NFS intensity measured in 16 bunch mode for 25 A (left

side) and 200 A (right side) pore sizes. The solid line shows the fit.

molecule. The decay rate gives information about the rate of relaxation which seems to

depend on the pore sizes. The smallest relaxation rate was obtained for the 25 A pores.

However, the relatively short experimental time window and the overlap of the signal from

subsequent bunches lead to a uncertainty of the relaxation rates.

In order to increase the sensitivity, the experiment was repeated in the single bunch

mode (time window ∼2.8 µs), allowing for a large observation window. Samples with

pore sizes 25 and 50 A have been used in this experiment in order to see the largest

deviation of the relaxation rate from that in the bulk. In order to get information about

the Lamb-Mossbauer factor and about the low-temperature behaviour of the quadrupole

splitting the temperature range has been extended to 28-207 K.

The obtained NFS spectra are shown in Fig. 7.2 for several temperatures. For all

temperatures except 28 K the time evolution of the intensity is well described by an

exponential decay modulated by a QB with a period of about 36 ns. The effect of multiple

scattering is limited to a weak acceleration of the natural decay, and the intensity is

well described by the quasi-kinematical approximation. The additional decay rate is

proportional to ξfLM and decreases with temperature due to the decrease of fLM . The

time spectra here differ strongly from those for the bulk (see Fig. 6.10, 6.11) where the

DB structure of multiple scattering is observed up to 190 K. The first minimum of the

DB structure for the 25 A and 50 A samples is at the same time of ∼400 ns. On the

other hand, the first minimum for the bulk sample at 50 K is at ∼20 ns. The position of


Figure 7.2: Time evolution of the NFS intensity measured in single bunch mode for

25 A (left side) and 50 A (right side) pore sizes. The solid line shows the fit.

this minimum is inverse proportional to ξfLM , therefore, ξfLM for the bulk sample is ∼20

times larger than for the nanoporous samples. Assuming roughly the same value of the

Lamb-Mossbauer factor, one can estimate that the effective thickness of the bulk sample

was ∼20 times larger than for the nanoporous samples.

The relaxation, which appears at temperatures above ∼ 160 K, is seen as an increase

of the decay rate. The non-exponential character of relaxation leads to a stretching of the

time spectra which is seen at highest temperatures.

The treatment of the data is performed here assuming an exponential relaxation func-

tion. In the quasi-kinematical approximation one can define the NFS intensity as

I(t) = I0(ξfLM)2e−(1+ζ)t/τ0 cos2(Ωet/2τ0) (7.1)

where the effective quadrupole splitting ~Ωe and the decay rate ζ is introduced. The

decay rate ζ is defined by the combination of the effective thickness parameter and the

relaxation rate

ζ =ξfLM

2+ 2λNFSτ0. (7.2)


0 40 80 120 160 2000.0

0.4

0.8

1.2 b

Dec

ay r

ate

ζ [

τ-1 0 ]

Temperature T [ K ]

100

101

102 a

Cou

ntra

te R

[ a

.u. ]

Figure 7.3: a) Temperature dependence of the countrate R for the 25 A (,4) and the

50 A (¥) samples. The data were scaled to the same value at 150 K. b) Temperature

dependence of the decay rate ζ (, ¤), of the effective thickness parameter ξfLM/2 (•, ¥)

and of the relaxation rate 2λNFS (⊕, ¢) for the 25 A and the 50 A samples, respectively.

The data were fitted with 3 independent parameters: I0, ~Ωe and ζ. The temperature

dependence of ζ for both nanoporous samples is shown in Fig. 7.3b. The data sets for both

25 A and 50 A samples almost coincide. ζ decreases with temperature up to 170 K and

increases above. Such a behaviour is explained by the opposite temperature dependences

of the two contributions in ζ. ξfLM/2 is the major contribution at low temperatures. It

decreases with temperature and its contribution at high temperatures becomes negligible.

On the other hand, 2λNFSτ0 is zero at low temperatures, but increases with temperature.

Its contribution to ζ is dominant at high temperatures.

The separation of these two contributions to ζ requires an additional independent

information. In order to obtain it we measured the countrate R. This parameter was

introduced in Section 6.4 and is the NFS intensity integrated over the experimental time

window observed during a short time of measurement. The temperature dependence of

the countrate is shown in Fig 7.3a. The connection between R and the NFS intensity I(t)

is given by

R = a′∫ ∞

20 ns

dt · I(t)/I0 = a(ξfLM)2

1 + ζe−0.14ζ (7.3)


Figure 7.4: Temperature dependence of the quadrupole splitting ~Ωe (a) and of the Lamb-

Mossbauer factor fLM (b) for the 25 A (¤), 50 A (♦) and bulk (•) samples.

where the lower integration limit, 20 ns, is the beginning of the experimental time win-

dow, a′ and a are proportionality coefficients. In the last expression ζ can be obtained

unambiguously from the NFS intensity and R is directly measured. The proportional-

ity coefficient a is obtained at 75 K, where ζ = ξfLM/2. By this procedure, we obtain

the effective thickness parameter ξfLM/2 which is shown as a function of temperature in

Fig. 7.3b. One can see that ζ and ξfLM/2 almost coincide up to ∼ 140 K. Above this

temperature the relaxation is activated and becomes visible in the experimental time win-

dow. Above 190 K the contribution of ξfLM/2 to ζ becomes negligible and the decay is

defined mainly by the relaxation. In the temperature range 150-190 K both contributions

to ζ are comparable. The correct value of 2λNFS here strongly depends on the correct

measurement of the countrate R.

7.4. Results 107

7.4 Results

The quadrupole splitting, the Lamb-Mossbauer factor and the relaxation rate are obtained

from the fit for both nanoporous samples.

The temperature dependence of the quadrupole splitting ~Ωe is shown in Fig. 7.4a.

The data sets for both nanoporous samples almost coincide. The quadrupole splitting

decreases linearly up to ∼160 K and falls down above. The low temperature linear de-

pendence can be approximated as

~Ωe ' 24.99(1 − 0.017 · T/Tg) (7.4)

where Tg = 179 K for DBP.

The quadrupole splitting observed in the bulk sample is also shown in the figure. It

is clearly seen that data sets behave differently at low temperatures. The value of ~Ωe

at T = 0 K differs by ∼2% for nanoporous and bulk samples. Also the dependence on

temperature below 160 K is more pronounced for the nanoporous samples. However, above

160 K all data sets almost coincide and show a characteristic non-linear decrease. This

can be identified with fast rotational dynamics in a restricted angular range which leads

to the same decrease of the quadrupole splitting for the bulk and nanoporous samples.

The temperature dependence of the Lamb-Mossbauer factor fLM is shown in Fig. 7.4b

for the nanoporous and the bulk samples. Absolute values of fLM for the nanoporous sam-

ples were obtained from ξfLM using the investigation of the same samples by NIS [ABvB+03],

where an absolute value of fLM has been obtained at 85 K. As a result we obtain that the

effective thickness ξ equals 3.0 for the 25 A sample and 3.2 for the 50 A sample. Using

this information we obtain fLM . The temperature dependence of the Lamb-Mossbauer

factor for both nanoporous samples is almost the same. The comparison with the bulk

sample shows that at low temperatures, where vibrations play a main role, the data sets

for bulk and nanoporous samples show the same temperature dependence. However, at

higher temperatures, the data sets deviate from each other. fLM for the bulk sample is

drastically decreased due to the softening of the glass former connected with fast β re-

laxation. On the other hand, fLM in nanoporous samples shows a similar behaviour, but

the decrease is much smaller. Thus softening also appears in the nanoporous samples but

with a much smaller amplitude. The relatively small decrease of fLM in the nanoporous

samples is also seen from the following argument. The maximum temperature where

experiments can be carried out is defined by a countrate ∼ 1 Hz. The countrate is pro-

portional to the square of ξfLM . For both, bulk and nanoporous samples, the maximum

temperature was roughly the same, 205 ÷ 210 K, which means that the values ξfLM are

the same. However, the effective thickness ξ for the bulk sample is at least 20 times larger

than that for the nanoporous samples. Therefore, the Lamb-Mossbauer factor has to be


Figure 7.5: Temperature dependence of the relaxation rate 2λNFS measured by NFS in the

25 A (¤) and 50 A (♦) samples and measured by NFS and MS in the bulk (•) sample. The

rotational relaxation rate λ2 measured by SRPAC in the bulk sample is also shown (F).

The lines show low temperature Arrhenius behaviour for the nanoporous (solid line) and

bulk (dashed line) samples.

20 times smaller. This ratio corresponds to the ratio of fLM for the nanoporous and the

bulk samples presented in the figure. Here one should also mention NIS measurements

of similar samples [ABvB+03]. The obtained partial density of states at the iron atom is

suppressed at low energies as compared to the bulk.

Fig. 7.5 shows the inverse temperature dependence of the relaxation rate 2λNFS for

the nanoporous and the bulk samples. Below 190 K both data sets follow an Arrhenius

dependence on temperature. The slope is less pronounced for the nanoporous samples.

The barrier energy ∆E of the Arrhenius law is ∼20% smaller for the nanoporous samples

as compared to the bulk sample. We know that below 190 K, relaxation observed by NFS

in the bulk sample is fully determined by rotational relaxation which follows an Arrhenius

dependence on temperature. It is natural to assume that also in the nanoporous samples

the Arrhenius behaviour is defined by rotational relaxation alone. This relaxation process

is identified with the slow β relaxation for the bulk sample. The same should be true for

the nanoporous samples.

Above 190 K 2λNFS observed in the bulk sample changes its behaviour and strongly

increases with temperature. Such behaviour is not observed for the nanoporous samples.

7.4. Results 109

Here, the relaxation rate follows the Arrhenius temperature dependence up to the maximal

measured temperature. Since the increase of 2λNFS for the bulk sample was identified

with structural α relaxation, we can say that this relaxational process is suppressed or

shifted to higher temperatures for the nanoporous samples.

As conclusion, we do not see any difference in the dynamics of glass former in pores of

25 A and 50 Adiameters. As compared to the bulk, the rotational dynamics seen by the

quadrupole splitting and by the relaxation rate is almost the same. On the other hand,

fast β relaxation seen by the Lamb-Mossbauer factor and structural relaxation seen as

the deviation of the relaxation rate from an Arrhenius temperature dependence is largely

suppressed for the nanoporous samples. We can say that the restriction of the volume

of the glass former leads to the suppression of the fast β relaxation and structural α

relaxation while slow β relaxation identified with molecular rotation remains almost the

same.

Chapter 8

Conclusion and Outlook

In this work a new method, Synchrotron Radiation based Perturbed Angular Correla-

tion (SRPAC), has been developed theoretically and successfully applied to study dy-

namics of a glass forming liquid.

The theoretical description of this method is based on the theory of angular correlation

which was worked out in the 60’s-70’s in application to the method of Time Differential

Perturbed Angular Correlation (TDPAC). The influence of the rotational motion on SR-

PAC has been studied theoretically in the stochastic Markov approximation of dynamics

and was expressed via rotational correlation functions C2(t) and C4(t). It is shown that

the full dynamical region can be separated into fast and slow relaxation regimes. In the

slow relaxation regime the relaxation is seen directly in time and both C2(t) and C4(t)

can be obtained from the data. The comparison of the characteristic rates of these two

correlation functions gives the possibility to extract information about the type of rota-

tion. In the fast relaxation regime only integrated information about the dynamics can

be obtained via the mean relaxation time of C2(t).

Both methods of nuclear resonant scattering, SRPAC and NFS, were applied to investi-

gate dynamical features of probe ferrocene molecules in the glass former dibutyl phthalate

for the transition from the glassy to the liquid state.

The rotational dynamics of the probes was measured by SRPAC in the frequency range

of 2·106÷4·1010 Hz. For the temperature range where both SRPAC and NFS are accessible

(0.9Tg−1.2Tg, Tg is the caloric glass transition temperature), a separation of the dynamics

into pure rotation and into translational motion of the center of mass has been performed.

The results show that below 1.1Tg these dynamical processes with characteristic times

longer than 100 ns decouple and exhibit a different dependence on temperature. The

rotational motion has the higher frequencies, it still exists when translational motion is

almost frozen out in the experimental time window.

Comparing these results to the data from dielectric spectroscopy, one finds that the

111

112 Chapter 8. Conclusion and Outlook

dynamics of the probe molecules reproduces the dynamics of the pure glass former. In

particular, the rotational dynamics of the probes slows down along the dielectric spec-

troscopy data at temperatures down to 1.3Tg. After a transition regime, it follows below

1.1Tg the slow β relaxation branch. On the other hand, the translational dynamics has

similarities with the structural α relaxation. Our results support the concept of decoupling

of the various types of motion below the critical temperature Tc as defined by mode cou-

pling theory and suggest an identification of the slow β relaxation with non-translational

molecular motion.

Together with slow dynamics seen directly in the experimental time window, fast lo-

calized motions were observed via the temperature dependence of the Lamb-Mossbauer

factor and of the quadrupole splitting. The excess decrease of the Lamb-Mossbauer fac-

tor and of the quadrupole splitting with increasing temperature was identified with the

appearance of localized relaxations which are interpreted as rattling and libration of the

probe molecule, respectively. The obtained mean square displacement and mean square

angular displacement show a similar behavior with temperature. Because of the charac-

teristic square-root dependences of these effects we attribute them to fast β relaxation.

NFS was applied also to study dynamics of a glass former confined in nanoporous

material with pores size of 25 and 50A. Again the glass former dibutyl phthalate doped

by ferrocene was chosen which gives the possibility to compare data with that obtained

in the bulk case. The result shows no particular dependence on the size of the pores. For

the used pore sizes the fast localized translational dynamics is suppressed and the fast

rotational dynamics is almost the same as in the bulk case. The characteristic rate of

the slow dynamics is similar to that obtained by SRPAC in the bulk sample and reduced

as compared to the relaxation rate obtained by NFS. All this leads to the conclusion

that the dynamics in nanopores is dominated by the rotational molecular motion and the

translational dynamics is suppressed as compared with the bulk.

In an outlook, several tasks can be considered. The theoretical model which was

used to explain the influence of dynamics on both NFS and SRPAC assumed a Markov

character of relaxation. Also, only one relaxation process was taken into account. It

would be interesting to consider how a more general model of relaxation would be seen

in NFS and SRPAC.

The theoretical analysis of SRPAC shows that one can extract both the characteristic

rate of relaxation and the model of rotation from the time evolution of the anisotropy.

However, the experimental statistics did not allow us to obtain information about the

model of rotation. This interesting task has to be solved in future.

The combination of NFS and SRPAC gives the unique possibility to obtain quan-

titative information about both the rotational and translational relaxation rates. The

values obtained in such way can be compared with the independently obtained transla-

113

tional relaxation rate from neutron scattering and rotational relaxation rate from NMR.

This allows to compare quantitatively the last two rates and gives important information

about their coupling. Due to the lack of neutron scattering data on dibutyl phthalate we

were not able to make this comparison in this work. In order to do so one can apply the

combination of NFS and SRPAC to other glass forming liquids like glycerol or o-terphenyl

which were investigated by many different techniques.

Possible applications of SRPAC are not restricted to soft condensed matter and to

the 57Fe isotope. The method can be used, in principle, as a high resolution spectroscopy

to investigate excited states of other isotopes. SRPAC becomes especially important in

the case of absence of or difficulties with the mother isotope, which makes impossible

to use traditional TDPAC, or in the case of a high transition energy which restricts the

use of NFS or MS to very low temperatures. In particular, the application of SRPAC

to 61Ni looks very promising. The appearance of nickel in many magnetic and biological

compounds makes it important to obtain information about the hyperfine interactions

experienced by the nuclei. First experiments which have been performed at the ESRF

Nuclear Resonant beamline on a nickel metal foil enriched in 61Ni demonstrates the prin-

cipal possibility to observe SRPAC on this isotope. The temperature dependence of the

measured magnetic hyperfine interaction reproduces the known decrease of the magne-

tization with temperature up to the Neel temperature. The relatively high countrate of

these measurements allows one to plan investigations of nickel compounds under special

conditions: on surfaces, under pressure, in biological compounds and etc.

114 Chapter 8. Conclusion and Outlook

Appendix A

Calculation of the NFS amplitude

As a basis for the calculation we use the expression obtained by Shvyd’ko [Shv99a] for

the NFS amplitude

E(t) =∞∑

k=1

(−1)k ξk

k!E(k)(t) (A.1)

where the multiple scattering amplitude of order k is formed from the self-correlation

function K(t) by the recursion relation

E(1)(t) =E0

τ0K(t) (A.2)

E(k+1)(t) =1

τ0

∫ t

0

dt · E(k)(t) ·K(t− t) (A.3)

This expression can be simplified by introducing K(t) and ψ(k)(t) as

K(t) = e−t/2τ0K(t) (A.4)

E(k+1)(t) =E0

τ0e−t/2τ0

(

t

τ0

)k1

k!ψ(k)(t) (A.5)

The recursion relation for ψ(k)(t) can be found by inserting eq. (A.5) into eq. (A.3)

E0

τ0e−t/2τ0

(

t

τ0

)k1

k!ψ(k)(t) =

=E0

τ0

1

τ0

∫ t

0

dt · e−t/2τ0

(

t

τ0

)k−11

(k − 1)!ψ(k−1)(t) · K(t− t)e−(t−t)/2τ0

(A.6)

ψ(k)(t) =

∫ t

0

dt

t

(

t

t

)k−1

k · ψ(k−1)(t) · K(t− t) =

=

∫ 1

0

d(

xk−1)

ψ(k−1)(tx) · K(t(1 − x))

(A.7)

115

116 Appendix A. Calculation of the NFS amplitude

Decreasing the index in eq. (A.1) by one we come to the following expression for the NFS

amplitude

E(t) = E0ξ

τ0· e−t/2τ0 · E(t) (A.8)

E(t) =∞∑

k=0

(−1)k (ξt/τ0)k

k!(k + 1)!ψ(k)(t) (A.9)

ψ(0)(t) = K(t) (A.10)

ψ(k+1)(t) =

∫ 1

0

d(

xk)

ψ(k)(tx) · K(t(1 − x)) (A.11)

Here E(t) reflects the influence of multiple scattering and relaxation on the NFS ampli-

tude.

The most simple case corresponds to K(t) = 1 (no diffusion and no hyperfine splitting).

Then ψ(k) = 1 for any k and the sum reduces to the well known expression [KAK79]

E(t) =J1(2

√

ξt/τ0)√

ξt/τ0≡ σ(ξt/τ0) (A.12)

where J1(x) is the Bessel function of the first kind and first order.

The second case which we consider here corresponds to the perturbation of the nuclear

state by the quadrupole interaction with K(t) = cos(Ωt/2). The calculation of ψ(k)(t) up

to k = 4 using the reccurent eq. (A.11) gives the following expression for E(t)

E(t) =∞∑

k=0

(−1)k (ξt/2τ0)k

k!(k + 1)!

(

ak(Ωt) · cos(Ωt/2) −ξ

2Ωτ0bk(Ωt) · sin(Ωt/2)

)

(A.13)

where

a0(x) = 1 b0(x) = 1

a1(x) = 1 b1(x) = 1

a2(x) = 1 b2(x) = 1 +2

x2

a3(x) = 1 − 12

x2b3(x) = 1 +

6

x2

a4(x) = 1 − 60

x2b4(x) = 1 +

8

x2+

48

x4(A.14)

The deviation of the coefficients ak(Ωt) and bk(Ωt) from 1 is inverse proportional to (Ωt)2.

If this value is large (large t and Ω not small) we can approximate ak(x) and bk(x) by 1,

which results in

E(t) = E0ξ

2τ0e−t/τ0 · σ(ξt/2τ0) · cos(Ωt/2 + ξ/4Ωτ0) (A.15)

117

At last, we consider a self-correlation function which corresponds to the unsplit state

and to the translational relaxation governed by the von Schweidler function

K(t) = 1 − (tλt)β (A.16)

In this case ψ(k)(t) can be presented as

ψ(k)(t) = 1 − (λtt)β Γ(1 + β)k!

Γ(k + β)(A.17)

Here we assume that λt is a small value and take into account only the first order of

(λtt)β. One can check the correctness of this expression inserting it to eq. (A.11)

ψ(k+1)(t) =

∫ 1

0

d(

xk)

·(

1 − (λtt)β Γ(1 + β)k!

Γ(k + β)· xk

)

·(

1 − (λtt)β(1 − x)β

)

'

' 1 − (λtt)β

(

Γ(1 + β)k!

Γ(k + β)

∫ 1

0

d(

xk)

xβ +

∫ 1

0

d(

xk)

(1 − x)β

)

=

= 1 − (λtt)β

(

Γ(1 + β)k!

Γ(k + β)

k

k + β+

Γ(1 + β)k!

Γ(1 + k + β)

)

=

= 1 − (λtt)β Γ(1 + β)(k + 1)!

Γ(k + 1 + β)

(A.18)

Inserting ψ(k)(t) into eq. (A.9) one obtains

E(t) = σ(ξt/τ0) − (λtt)βΓ(1 + β)

∞∑

k=0

(−1)k (ξt/τ0)k

Γ(k + β)(k + 1)!=

= σ(ξt/τ0) − (λtt)βσβ(ξt/τ0)

(A.19)

where the generalized σx(z) is introduced as

σx(z) = Γ(1 + x)∞∑

k=0

(−1)k zk

Γ(k + x)(k + 1)!(A.20)

For x = 1 this function reduces to σ(z).

Let ti be the roots of E(t) in the static case λt = 0, i.e. σ(ξti/τ0) = 0. One can find

the shifts ∆ti of the roots due to relaxation from an equation

E(ti + ∆ti) = 0 (A.21)

We assume that ∆ti is small, then we can expand the last expression in a Taylor series.

E(ti) + ∆tiE′(ti) = 0 (A.22)

∆tiσ′(ξti/τ0) − (λtti)

βσβ(ξti/τ0) = 0 (A.23)

Straightforward one can obtain an expression for the relative shift ∆ti/ti as a function of

the relaxation rate

∆titi

= (λtti)βCi(β) (A.24)

Ci(β) =σβ(ξti/τ0)

ξti/τ0 · σ′(ξti/τ0)(A.25)

118 Appendix A. Calculation of the NFS amplitude

Appendix B

Electric quadrupole interaction

The coupling between nucleus and environment is described by the electrostatic inter-

action of the nuclear charge and the electric and magnetic fields on the nucleus formed

by its surroundings (electrons, crystal structure). The interaction Hamiltonian H can be

described by [Kop94]

H = HC +HQ +HM (B.1)

where HC refers to the Coulomb interaction between the nucleus and the electron density

at the nuclear site, HQ is the interaction between the nuclear quadrupole moment and

the electric field gradient at the nucleus (electric quadrupole interaction), and HM is the

interaction between the nuclear magnetic dipole moment and the effective magnetic field

at the nucleus (magnetic dipole interaction). Interactions of higher order of multipole

expansion can be neglected because their energies are by several orders of magnitude

smaller [Kop94].

If the distribution of the nuclear charge deviates from spherical symmetry then the

nucleus has a quadrupole moment eQij [GLT78], which is a tensor of second order. If the

nuclear charge distribution has at least an axial symmetry then only one component of this

tensor is nonzero. The quadrupole moment Q is positive for an elongated (cigar-shaped)

nucleus, and negative for a flattened (pancake-shaped) nucleus. A quadrupole moment

different from zero exists only for nuclear states with spin quantum number I > 1/2. The

quadrupole moment interacts with a non-spherical electronic charge distribution at the

nuclear site. The non-sphericity can be described by the electric field gradient (EFG)

tensor, whose components are defined as the second derivatives of the electric potential

V produced by extra-nuclear charges at the nuclear site (Vij = ∂2V/∂xi∂xj). In contrast

to nuclear charge distribution, the electronic charge distribution is generally not axially

symmetric. It is nevertheless possible to find a principal axis coordinate system, where all

non-diagonal components of Vij vanish. Due to the Laplace’s equation (Vxx + Vyy + Vzz =

0) and using the assumption | Vzz |>| Vxx |>| Vyy | one can choose two independent

119

120 Appendix B. Electric quadrupole interaction

parameters to describe the EFG: q ≡ Vzz/e and the asymmetry parameter η defined as

η = (Vxx−Vyy)/Vzz. The interaction Hamiltonian between the quadrupole moment of the

nucleus in state I and the EFG at the site of the nucleus is

HQ =e2Qq

4I(2I − 1)

(

3I2z − I(I + 1) + η

(

I2x − I2

y

))

(B.2)

where hats over spins denote the operator nature of these values. Eigenvalues for this

Hamiltonian in the case of I = 3/2, that is the value of spin for the excited state of 57Fe,

are

EQ =e2Qq

12

(

3m2I −

15

4

)

√

1 + η2/3 (B.3)

where mI = I, I − 1, . . . ,−I is the magnetic quantum number (projection of the spin

to the z axis of the EFG coordinate system). The electric quadrupole interaction causes

a splitting of the (2I + 1)-fold degenerate energy level of a nuclear state into sub-levels

characterized by the magnitude of the magnetic quantum number | mI |. These levels can

not be distinguished by the sign of the magnetic quantum number because of the presence

of mI to the second power in eq. (B.3), they are therefore always twofold degenerate. In

the case of 57Fe the ground state (I = 1/2) is not split by the EFG and the excited state

(I = 3/2) splits into two twofold degenerate sublevels with mI = ±1/2 and mI = ±3/2.

It is convenient to introduce the angular frequency Ω that is equivalent to the energy

splitting between these states and also called the quadrupole splitting:

~Ω = e2Qq/2 (B.4)

where we assume that the EFG is axially symmetric (η = 0).

As the nuclear quadrupole moment is constant for each nuclear level, changes of the

quadrupole splitting observed for the same compound under different experimental condi-

tions result from changes of the EFG at the nucleus. One can consider two sources which

contribute to the EFG [GLT78]: charges on distant ions which surround the Mossbauer

atom in a non-cubic symmetry, called lattice contribution, and anisotropic electron dis-

tribution in the valence shell of the Mossbauer atom, called valence electron contribution.

Appendix C

Calculation of the SRPAC intensity

The probability of the incoherent scattering of a SR pulse is proportional to the angular

correlation function for two successive radiations emitted from an initial random state

with the perturbing interactions acting on the intermediate state. An expression for the

general form of this function was derived by R. M. Steffen and K. Alder (see eq. (13.137)

in [SA75]).

For the incoherent NRS of SR we can make some assumptions. First, we assume that

the multipolarity of the photon which accompanies to the nuclear transitions is fixed to one

value (M1 for the 1/2 → 3/2 transition of 57Fe). Second, the incoming beam is assumed

to be polarized in the horizontal plane, whereas the polarization of the scattered photon is

not observed. Third, we assume that the perturbing interactions on the ensemble of nuclei

are isotropic, i.e. the overall effect on the ensemble is such that it does not introduce a

privileged direction in space.

Using eqs. (13.137,12.293,13.107) of [SA75], the probability of incoherent NRS of the

incoming photon with direction kin and polarization σin scattered into the direction kout

and divergence dΩout is:

dW (kinσin,kout; t) =dΩout

4π

∑

l,q,q′

B∗lq(γin, σin) ·D(l)

qq′((kinσin → Sk)

·Gll(t) · Al(γout) ·D(l)q′0(Sk → kout)

(C.1)

Here Gll(t) is a perturbation factor which completely describes the dynamics of the nu-

cleus, D(l)qq′(kinσin → Sk) is the matrix element of the rotation from the coordinate system

connected with the incoming photon (kin - Z-axis and σin - X-axis) to the coordinate sys-

tem connected with nucleus (Sk), D(l)q′0(Sk → kout) is the matrix element of the rotation

from the coordinate system Sk to the coordinate system connected with the outcoming

photon (kout - Z-axis), and coefficients A and B will be defined below. This expression

121

122 Appendix C. Calculation of the SRPAC intensity

can be simplified further to


4π

∑

l,q

B∗lq(γin, σin)Al(γout)Gll(t) ·

√

4π

2l + 1Y ∗

lq(φ′, ϑ′) (C.2)

where we use the relation:

∑

q

D(l)qq′(Ω1)D

(l)q′0(Ω2) = D

(l)q0 (Ω1Ω2) =

√

4π

2l + 1Y ∗

lq(φ′, ϑ′) (C.3)

Here φ′ and ϑ′ are the angles that define the direction of the scattered beam in the

coordinate system connected with the incoming beam. The coefficients Blq(γ, σ) and

Al(γ) are (see eqs. (12.232) and(12.182) in [SA75]):

Bl0(γ, σ) = Al(γ) = Fl(L,L, Ig, Ie) (C.4)

Bl±1(γ, σ) = 0 (C.5)

Bl±2(γ, σ) =1

4

√

(l + 2)!

(l − 2)!Fl(L,L, Ig, Ie) (C.6)

where L is the multipolarity of radiation, Ig and Ie are the spins of the ground and excited

states, and Fl(L,L, Ig, Ie) is given by eq. (12.168) in[SA75]

Fl(L,L, Ig, Ie) = (−1)Ig+Ie−1(2L+ 1)√

(2l + 1)(2Ie + 1)

(

L L l

1 − 1 0

)

L L l

Ie Ie Ig

(C.7)

The sum over q in eq. (C.2) can be taken in the following way:

√

4π

2l + 1

∑

q

Blq · Y ∗lq(φ

′, ϑ′) =

√

4π

2l + 1·(

Bl0 · Y ∗l0(φ

′, ϑ′) +Bl2 · Y ∗l2(φ

′, ϑ′) +Bl−2 · Y ∗l−2(φ

′, ϑ′))

=

Bl0Pl(cosϑ′) +

√

4π

2l + 1Bl2 · 2Re(Yl2(φ

′, ϑ′)) =

Al

(

Pl(cosϑ′) +

1

2Pl2(cosϑ

′) · cos 2φ′

)

(C.8)

where Pl(x) is a Legendre polynomial and Pl2(x) is an associated Legendre polyno-

mial [AS70]. Using this relation the probability of the scattering can be written as


4π

∑

l

A2l (γ) ·Gll(t) ·

(

Pl(cosϑ′) +

1

2Pl2(cosϑ

′) · cos 2φ′

)

(C.9)

where the coefficient l varies in the region 0 6 l 6 min(2L, 2Ie).

For many isotopes (57Fe, 119Sn, 61Ni) the multipolarity L equals 1. Then, the sum

in eq. (C.9) is restricted by two terms with l = 0 and 2. The first term equals 1. The

123

term for l = 2 simplifies turning to another coordinate system, where the z-axis coincides

with the vertical direction. We define ϑ and φ as a polar and azimuthal angles in this

coordinate system. Then

P2(cosϑ′) +

1

2P22(cosϑ

′) · cos 2φ′ = −2P2(cosϑ) (C.10)

Finally the angular probability can be written as

dW (ϑ; t)

dϑdφ=

1

4π

(

1 − 2A22(γ)P2(cosϑ)G22(t)

)

(C.11)

The result obtained for SRPAC can be compared with the angular probability for the

directional TDPAC, which for L = 1 can be written as [SA75]

dW (ϑ′; t)

dϑ′dφ′=

1

4π(1 + A2(γ1)A2(γ2)P2(cosϑ

′)G22(t)) (C.12)

In application to the 57Fe isotope the spin transition 1/2 → 3/2 is observed by SRPAC

and a cascade of spins 5/2 → 3/2 → 1/2 is observed by TDPAC. The values of the

anisotropy coefficients are A2(γ) = 1/2 for SRPAC and A2(γ1) = 1/10, A2(γ2) = 1/2 for

TDPAC. The corresponding angular probabilities reduce to

dW (ϑ; t)

dϑdφ=

1

4π

(

1 − 1

2P2(cosϑ)G22(t)

)

(C.13)

for SRPAC anddW (ϑ′; t)

dϑ′dφ′=

1

4π

(

1 +1

20P2(cosϑ

′)G22(t)

)

(C.14)

for TDPAC. One can see that the coefficient before the perturbation factor is 10 times

larger for the SRPAC. This difference partly appears due to the polarization and partly

due to the low spin of the ground state.

124 Appendix C. Calculation of the SRPAC intensity

Appendix D

Calculation of G22(t) by the

eigensystem method

The time evolution of the nuclear state driven by quadrupole interaction and stochastic

reorientation is described by eq. (4.10). Additionally one needs to introduce a representa-

tion of the nuclear and stochastic states in order to get an expression for the perturbation

factor. It was found by Winkler [Win76] that the coupled representation | x(Ff, Jk;κ))is quite convenient for this purpose:

| x(Ff, Jk;κ)) =∑

Nµ

〈JµkN | Ff〉 | J, µκ) | kN) (D.1)

where angular brackets denote Clebsch-Gordan coefficient, | J, µκ) describes the molecular

orientation in angular momentum space and | kN) is the density matrix of the nuclear

state in the spherical tensor representation (for an explanation of this representation

see [Blu81]).

Then the perturbation factor is given by projection of the operator U(t) to the element

|X(k0, 0k, 0)) of the X representation (see eqs.(42) and(44) in [Win76])

Gkk(t) = (x(k0, 0k, 0)| exp

[(

− i

~H× + R

)

t

]

|x(k0, 0k, 0)) (D.2)

Using information about the projection of the operator R to | J, µκ) (see eq. (4.17))

one can rewrite eq. (34) in [Win76] as

(x(F2f2, J2k2;κ2)|R|x(F1f1, J1k1;κ1)) = −λJ1 · δJ1J2δf1f2δF1F2δk1k2δκ1κ2 (D.3)

and for the Liouville operator of the quadrupole interaction (eqs. (39) and (40) in [Win76])

(x(F2f2, J2k2;κ2)|H×|x(F1f1, J1k1;κ1) = δF1F2δf1f2δκ1κ2 · ~Ω√

5 ·· (−1)F1+κ1+1 ·

(

(−1)k1+k2 − 1)

·√

(2J1 + 1)(2k1 + 1)(2J2 + 1)(2k2 + 1) ·

·(

J1 J2 2

−κ1 κ1 0

)

·

k1 k2 2

J2 J1 F1

·

k1 k2 2

3/2 3/2 3/2

(D.4)

125

126 Appendix D. Calculation of G22(t) by the eigensystem method

where figure brackets denote a 6j-symbol and, as compared with [Win76], we take into

account the quadrupole character of the hyperfine interaction.

The next step is to build the matrix A

A = − i

~H× + ~R (D.5)

in the x-representation. To decrease the dimension of this matrix one can use the property

that only states which couple with the state |x(k0, 0k; 0)) have to be taken into account

where k = 2 since we are interested in the calculation of G22(t). It follows immediately

that F1 = F2 = 2, f1 = f2 = 0 and κ1 = κ2 = 0. Looking to the indexes J and k we

obtain that only 5 states are coupled with J, k = 0, 2. Therefore the matrix A is a

6× 6 matrix in the representation J, k : 0, 2, 2, 1, 2, 2, 2, 3, 4, 2, 4, 3. Below

we give a precise expression for this matrix.

A = ~Ω ·

0 i√

625

0 −i√

1425

0 0

i√

625

− λ2

~Ω−i√

335

0 −i√

48175

0

0 −i√

335

− λ2

~Ω−i√

64245

0 −i√

1849

−i√

1425

0 −i√

64245

− λ2

~Ωi 335

0

0 −i√

48175

0 i 335

− qλ2

~Ω−i√

1049

0 0 −i√

1849

0 −i√

1049

− qλ2

~Ω

(D.6)

Here we introduce the parameter q = λ4/λ2.

The next step will be to diagonalize this matrix, i.e. to find a set of eigenvalues ε(j)

and normalized eigenvectors ~v(i) which solve the equation

A~v(j) = ε(j)~v(j) (D.7)

After this set is found the perturbation factor is calculated as

G22(t) =6∑

j=1

|v(j)0 |2eε(j)t (D.8)

In Fig. D.1 we give a program code written in “Mathematica-2.4” software which

calculates G22(t)

The presented method gives the numerical solution of the perturbation factor for any

values of the parameters of the relaxation λ2 and q. An analytical solution can not be

obtained in the general case. However, an analytical approximation of the perturbation

factor for the case of a small value of the relaxation rate λ2 can be found.

The set of eigenvalues ε(i) is the solution of the equation

det|A− ε1| = 0 (D.9)

127

Simulation of the perturbation factor G22 HtLInitial parameters of the model are Λ2, q and W.

W = 24; Λ2 = 5; q = 3;

Definition of the matrix A

A = W *NA

i

k

jjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjj

0 15

ä!!!!6 0 - 1

5ä

!!!!!!!14 0 0

15

ä!!!!6 -Λ2

W-ä

"########335

0 - 45

ä"######3

70

0 -ä"########3

35-Λ2

W- 8 ä

7!!!!!5

0 - 37

ä!!!!2

- 15

ä!!!!!!!14 0 - 8 ä

7!!!!!5

-Λ2W

3 ä35

0

0 - 45

ä"######3

70 3 ä

35-Λ2 q

W- 1

7ä

!!!!!!!10

0 0 - 37

ä!!!!2 0 - 1

7ä

!!!!!!!10 -Λ2 q

W

y

zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzE;

Solution of the eigensystem problem

8Ε, v< = Eigensystem@AD;Normalization of the vectors v and extracting É v1

HjL É2u = TableA vPj,1T2

Úi=16 vPj,iT2 , 8j, 1, 6<E;

Calculation of the perturbation factor

G@t_D = ReAâj=1

6

uPjT * ãΕPjT tE;Simulation of the perturbation factor and saving data

DataSim = Table@8t, G@tD<, 8t, 0, 3, 0.01<D;ListPlot@DataSim, PlotJoined ® True, PlotRange ® AllDSetDirectory@".sciencesrpacTheory"D;Export@"g-rdm-d2-Q24.dat", DataSim, "Table"D;

0.5 1 1.5 2 2.5 3

-0.4

-0.2

0.2

0.4

0.6

0.8

1

Graphics

Figure D.1: “Mathematica-2.4” program code for the simulation of the perturbation

factor.


which is an equation of 6-th power and can not be solved analytically in general. For the

static case when λ2 = 0, this equation degenerates and gives the roots:

ε(1)0 = ε

(4)0 = 0

ε(2)0 = ε

(5)0 = iΩ (D.10)

ε(3)0 = ε

(6)0 = −iΩ

which correspond to the exponent indexes of the statical perturbation factor

G22(t) =1

5· e0·t +

2

5

(

eiΩt + e−iΩt)

(D.11)

In order to solve eq. (D.9) for small λ2 we consider the roots in the shape of a Taylor

series

ε(i) = ε(i)0 + a(i)λ2 + b(i)λ2

2 (D.12)

up to the second order of λ2. Then eq. (D.9) has to be equivalent to∏6

i=1(ε− ε(i)). The

equivalence implies that the coefficients of any power of ε have to be the same up to

the second order of λ2. The obtained system of equations can be solved and gives the

following roots (up to the first order of λ2)

ε(1) = − 2

35(5 + 9q)λ2 ε(4) = −1

7(4 + 3q)λ2 (D.13)

ε(2) = iΩ − (A−B)λ2 ε(5) = iΩ − (A+B)λ2 (D.14)

ε(3) = −iΩ − (A−B)λ2 ε(6) = −iΩ − (A+B)λ2 (D.15)

where

A = (75 + 37q)/140 (D.16)

D =√

1009q2 − 450q + 225/140 (D.17)

Six roots appear in this common solution which is valid for any q. However, looking to the

analytical solution for the SCM with q = 1 (see eq. (E.17)) we can conclude that only three

roots are physical and have to be taken into account. In the SCM the eigenvalues are:

−4λ2/5 and ±iΩ − 3λ2/5. Comparing these values with ε(1)-ε(6) at q = 1 we obtain that

ε(1), ε(2) and ε(3) have to be taken as exponent indexes. Using the obtained eigenvalues

one can find the deviation of the coefficients v(j)0 from the static case using eq. (D.7). Here

we will not do it and use the static values. Then the perturbation factor can be written

as

G22(t) =1

5e−pht +

4

5e−pot cos(Ωe − φ) (D.18)

ph = λ22

35(5 + 9q) (D.19)

po = λ275 + 37q −

√

1009q2 − 450q + 225

140(D.20)

129

The shift of the precession frequency Ω to Ωe and phase shift φ are not calculated precisely

here. The solution of eq. (D.9) up to the third order of λ2 is necessary for this purpose.

Appendix E

Calculation of G22(t) by the resolvent

method

The calculation of the perturbation factor in the SCM can be done in the resolvent

formalism that was developed by Blume and Dattagupta in application to TDPAC and

MS [Dat87, Dat81].

This method considers the Laplace image of the perturbation factor

G22(λ) =

∫ ∞

0

dtG22(t)e−λt (E.1)

In the static case the Laplace transformation of eq. (4.6) leads to

G022(λ) =

1

5

(

1

λ+

4λ

λ2 + Ω2

)

(E.2)

The theory predicts that the modification of the perturbation factor due to relaxation can

be expressed in the frame of SCM as (see eq. (4.27) in [Dat81])

G22(λ) =G0

2(λ+ λ2)

1 − λ2G02(λ+ λ2)

(E.3)

where λ2 is the SCM jump rate introduced in eq. (4.16). After inserting the expression

for the static perturbation factor this equation can be written as

G22(p) =1

Ω

p2 + 2pp2 + p22 + 1/5

p3 + 2p2p2 + p(p22 + 1) + 4p2/5

(E.4)

where p = λ/Ω and p2 = λ2/Ω. To make the inverse Laplace transform to the time

variable one has to expand this expression in simple fractions and has therefore to find

the roots of the denominator. This can be performed analytically for an equation of the

third power. Notice that for TDPAC and SRPAC the power of the denominator is equal

2L where L is the spin of the excited state. L = 3/2 in the case of 57Fe and L = 5/2 for

131

132 Appendix E. Calculation of G22(t) by the resolvent method

the usual TDPAC isotopes. This leads to the third and fifth power of the denominator,

correspondingly. In the TDPAC case only a numerical solution is possible [Dat81].

Expanding eq. (E.4) and making the inverse Laplace transformation we get the ex-

pression for the perturbation factor in the time domain

G22(t) = Ae−αΩt +Be−βΩt cos(γΩt) + Ce−βΩt sin(γΩt) (E.5)

where

α =2

3p2 − s (E.6)

β =2

3p2 +

1

2s (E.7)

γ =

√3

2r (E.8)

A =4

135

9 + 45s2 + 30sp2 + 5p22

r2 + 3s2(E.9)

B =1

135

135r2 + 225s2 − 120sp2 − 4(9 + 5p22)

r2 + 3s2(E.10)

C =

√3

135

−45s3 + 60r2p2 + 60s2p2 + s(45r2 − 4(9 + 5p22))

r(r2 + 3s2)(E.11)

s = u+ v (E.12)

r = u− v (E.13)

u =

(

p32

27− p2

15+ w

)1/3

(E.14)

v =

(

p32

27− p2

15− w

)1/3

(E.15)

w =

√

5p42 − 22p2

2 + 25

675(E.16)

The dependence of the coefficients α, β, γ and A, B, C on λ2/Ω is shown in the Fig. E.1.

This rather complicated expression for G22(t) strongly simplified in the case of slow

and fast relaxation.

The slow relaxation regime is defined by λ2/Ω << 1. Then, up to the second order of

λ2/Ω, the perturbation factor can be written as

G22(t) ' 1

5e−4λ2t/5 +

4

5e−3λ2t/5 cos(Ωet− φ) (E.17)

Ωe = Ω

(

1 − 4λ22

25Ω2

)

(E.18)

φ =4λ2

5Ω(E.19)

The fast relaxation regime is defined by λ2/Ω >> 1 and, up to the first order of Ω/λ2,

133

Figure E.1: Dependence of the coefficients A, B, C and α, β, γ in eq. (E.5) on λ2/Ω.

the perturbation factor is expressed as

G22(t) ' exp

(

−4Ω2

5λ2

t

)

(E.20)

134 Appendix E. Calculation of G22(t) by the resolvent method

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Acknowledgments

I thank all persons who have supported me during this thesis work.

I would like first of all to thank Prof. Winfried Petry who made this thesis possible.

He has invited me to the TU Munchen, gave me the freedom to develop my own ideas

and was ready always to discuss them.

I am very grateful to Dr. Uwe van Burck who supervised me during the last two years

of my thesis. He made a huge contribution to this work. A lot of questions was solved

during our discussions and I really enjoyed the teamwork with him. I also would like to

thank him for the enormous job in correction of this thesis.

I thank Dr. Sasha Chumakov who also strongly contributed to this thesis. The

experiments presented here would have been impossible without his help. Also, many of

the ideas presented here were born during our ”fresh air” brainstorms.

I would like to thank Dr. Tanja Asthalter, who supervised me during the first two

years of my PhD study and introduced me into the technical and operational aspects of

the beamline. Also I thank her for the reading and correction of this thesis.

I want to express my gratitude to Dr. Rudolf Ruffer, who hosted me in his group at the

ESRF and provided in-house beamtime to perform the experiments. Also I appreciated

the beautiful atmosphere in the group created by Dr. Alessandro Barla, Dr. Bryan Doyle,

Dr. Hans-Christian Wille, Dr. Helge Thieß, Thomas Roth, Jean-Philippe Celse and other

former and present colleagues. Especially, I would like to thank Dr. Olaf Leupold for many

discussions and for help at any time.

I thank Dr. Hermann Franz for the possibility to perfom experiments at Petra I and

for many useful remarks. I also would like to thank Kirill Messel and Gerd Wellenreuther

who participated in some of the experiments presented here.

Special thanks I would like to give to Prof. Gennadii V. Smirnov and Dr. Valentin

G. Semenov who helped me to come to Munich and were always ready for discussion and

help.

Several discussions with Prof. S. Dattagupta have been useful to clarify theoretical

aspects of this work. I also thank him for the preliminary reading of this manuscript.

Thanks to Dr. P. Harter, Dr. H. Schottenberger and Dr. A. Huwe for the preparation

143

144 Bibliography

of the samples.

I would like also to thank my colleagues in E13, especially Dr. Nils Wiele, Dr. Walter

Schirmacher, Dr. Joachim Wuttke and Dr. Andreas Meyer for interesting and useful

discussions.

I would like to thank Edith Lubitz and Elke Fehsenfeld who helped me to s olve many

administrative problems.

I would like to thank my family and especially my father. His words that Mossbauer

spectroscopy is impossible in liquids became the starting point of this work.

Finally thanks to Anja for the encouragement and patience, especially during the last

few months of this work.

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